Section Formula Calculator (Internal and External Division, Midpoint and Centroid)

Choose the kind of point and the dimension, then enter the coordinates of the two points A and B and the ratio m:n. The formula below is linked to the input fields, so you can also edit the coordinates or the ratio right in the formula.

Point to find

Dimension

Point A x-coordinate x₁

Point A y-coordinate y₁

Point A z-coordinate z₁

Point B x-coordinate x₂

Point B y-coordinate y₂

Point B z-coordinate z₂

Point C x-coordinate x₃

Point C y-coordinate y₃

Point C z-coordinate z₃

Ratio part m (from A to the point)

Ratio part n (from the point to B)

You can use decimals, negative numbers and fractions such as 3/4. A blank coordinate counts as 0, and a blank ratio number counts as 1. The ratio m:n is in the order "m from A, then n from there to B".

Result and figure

Enter the coordinates of the two points and the ratio in the fields on the left and press "Calculate". The result and a figure will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the coordinates of two points \(\mathrm{A}\) and \(\mathrm{B}\) and a ratio \(m:n\), and you get the coordinates of the point that divides segment \(\mathrm{AB}\) in the ratio \(m:n\), internally or externally, on the spot
The midpoint (internal division in the ratio \(1:1\)) and the centroid of a triangle (the average of the coordinates of its three vertices) can be found on the same page
Works on the number line (1D), in the coordinate plane (2D) and in space (3D)
The answer is shown both as an exact fraction in lowest terms, such as \(\dfrac{13}{2}\), and as a decimal. You can also see the steps with the numbers put into the formula
A figure shows where the point is on the segment (inside it, or outside on which side). Coordinates and ratios can be decimals, negative numbers or fractions such as 3/4
An explanation of the "criss-cross" in the formula (why \(m\) and \(n\) seem to swap places) and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page

Enter the ratio \(m:n\) in this order - \(m\) counted from \(\mathrm{A}\), then \(n\) from there to \(\mathrm{B}\). Swapping the order gives a different point.

What is this calculation used for?

Moving smoothly between two points in animation and games (computer graphics and video)

When a character or a camera moves from point A to point B, the computer finds its position in every frame as "the point dividing segment AB internally in the ratio \(t : \left(1-t\right)\)" (where \(t\) is a number between \(0\) and \(1\)). This is called linear interpolation (lerp), the most basic calculation in computer graphics.
Increase t little by little from 0 to 1, and the object moves from A to B at a steady speed; change how fast t increases, and you get natural motion that starts slowly and stops slowly. Blending colors, resizing images and morphing between two faces all apply the same internal division calculation to colors or coordinates.

Balancing a seesaw or a balance scale (physics and center of gravity)

Put a 1 lb weight and a 2 lb weight on the two ends of a 3 ft rod (assume the rod's own weight is small enough to ignore). Where should the support go so that it balances? That is exactly an internal division calculation. The answer is 2 ft from the 1 lb end: the center of gravity is an average weighted by the weights, and it moves toward the heavier end. As a formula, with mass \(m_1\) at position \(x_1\) and mass \(m_2\) at position \(x_2\), the center of gravity is \(\dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\), the same form as the internal division formula.
Here it does not look like "the coordinate of the lighter end is multiplied by the heavier mass", because the masses themselves are the weights. In internal division, which works with the ratio of lengths, the weight is "the length of the part on the other side". The same idea is used to find the center of gravity of triangles, plates, cars and airplanes.

Combining class averages weighted by the number of students (statistics and weighted averages)

If a class of 40 students averages 60 points and a class of 20 students averages 90 points, the combined average of the two classes is not 75 but \(\dfrac{40 \times 60 + 20 \times 90}{40 + 20} = 70\) points. Think of 60 and 90 as two points \(\mathrm{A}\) and \(\mathrm{B}\) on the number line: the answer divides segment \(\mathrm{AB}\) internally in the ratio \(20:40\), which is the ratio of students \(40:20\) swapped around. The answer moves toward 60, the side with more students, because the other side's number of students is the weight. It is exactly the same swap as the criss-cross on this page.
Price indexes, batting averages, overall survey ratings and the average cost per share of an investment are all averages that combine groups of different sizes, and all take this form. It keeps you from the mistake of simply adding and dividing by 2, which gives too much weight to the smaller group.

Placing stakes between two points in surveying and civil engineering (construction)

In road and site construction, workers use the coordinates of two surveyed points to find "the position a set fraction of the way between them" and drive a stake there. The points that split a 400 ft stretch into four equal parts, at 100, 200 and 300 ft, are the internal division points for the ratios 1:3, 1:1 and 3:1.
With 3D coordinates that include height, the same formula finds points partway up a slope. Even for a tunnel dug from both ends that must meet exactly in the middle, many coordinate calculations like this keep the work accurate.

Finding a point along the way or a halfway meeting place in map apps (geographic information)

Treat the latitude and longitude of two places as coordinates, and the midpoint formula gives a place about halfway between them. This is used by features that suggest meeting places, and to split a long route into evenly spaced points to check the elevation or the weather along the way.
Over long distances you need calculations that take the round shape of the Earth into account, but within a city, treating the coordinates as flat and using internal division makes almost no difference in practice.

Formulas and figures

Internal division (the point dividing a segment internally in the ratio \(m:n\))

Figure

Standard notation (the usual math form)

\(x\) \(=\) \((\) \(n\) \(\times\) \(x_1\) \(+\) \(m\) \(\times\) \(x_2\) \()\) \(\div\) \(m+n\)

In words (symbols replaced with words)

⑥ \(x\): x-coordinate of the point \(\mathrm{P}\) \(=\) \((\) ① \(n\): ratio part from the point to \(\mathrm{B}\) \(\times\) ② \(x_1\): x-coordinate of \(\mathrm{A}\) \(+\) ③ \(m\): ratio part from \(\mathrm{A}\) to the point \(\times\) ④ \(x_2\): x-coordinate of \(\mathrm{B}\) \()\) \(\div\) ⑤ \(m+n\): total of the ratio

The formula in words

① Let \(\mathrm{P}\) be the point dividing segment \(\mathrm{AB}\) internally in the ratio \(m:n\). Take the \(n\): ratio part from the point to \(\mathrm{B}\)

② and multiply it by the \(x_1\): x-coordinate of point \(\mathrm{A}\)

③ Take the \(m\): ratio part from \(\mathrm{A}\) to the point

④ and multiply it by the \(x_2\): x-coordinate of point \(\mathrm{B}\) (then add the two products)

⑤ Divide the sum by the \(m+n\): total of the ratio

⑥ and you get the x-coordinate of the point \(\mathrm{P}\) (the \(y\)- and \(z\)-coordinates are found with exactly the same formula)

Quick example

The \(x\)-coordinate of the point \(\mathrm{P}\) dividing \(\mathrm{A}(1,\ 2)\), \(\mathrm{B}(7,\ 5)\) internally in the ratio \(2:1\) is

x-coordinate of \(\mathrm{P}\) \(=\) \((\) ratio part on the \(\mathrm{B}\) side (1) \(\times\) x-coordinate of \(\mathrm{A}\) (1) \(+\) ratio part on the \(\mathrm{A}\) side (2) \(\times\) x-coordinate of \(\mathrm{B}\) (7) \()\) \(\div\) total of the ratio (3)

\(x = \dfrac{1 \times 1 + 2 \times 7}{2 + 1} = \dfrac{15}{3} = 5\)

\(y = \dfrac{1 \times 2 + 2 \times 5}{2 + 1} = \dfrac{12}{3} = 4 \quad \Rightarrow \quad \mathrm{P}(5,\ 4)\)

Key idea

We divide in the ratio \(m:n\), so why is \(m\) multiplied by \(x_2\) (the coordinate of point B) and \(n\) by \(x_1\) (the coordinate of point A)? This swap, the "criss-cross", is the main reason the section formula is hard to remember. If you follow the reasoning, you will see that the swap is the natural result. The point \(\mathrm{P}\) lies \(\dfrac{m}{m+n}\) of the way from \(\mathrm{A}\) to \(\mathrm{B}\). \(\mathrm{AP} : \mathrm{PB} = m : n\) says that the whole segment \(\mathrm{AB}\) is split into \(m + n\) equal parts, and \(m\) of them are on the \(\mathrm{A}\) side. So start at the coordinate of \(\mathrm{A}\) and move \(\dfrac{m}{m+n}\) of the change from \(\mathrm{A}\) to \(\mathrm{B}\), \(x_2 - x_1\). Simplifying this gives \(x = x_1 + \dfrac{m}{m+n}\left(x_2 - x_1\right) = \dfrac{\left(m+n\right)x_1 + m x_2 - m x_1}{m+n} = \dfrac{n x_1 + m x_2}{m+n}\) and only \(n\) is left on the \(x_1\) side. To remember it by its meaning: "the number for the far side is the weight". When \(\mathrm{P}\) is close to \(\mathrm{A}\), the \(m\) on the \(\mathrm{A}\) side is small and the \(n\) on the \(\mathrm{B}\) side is large. The answer should be close to the coordinate of \(\mathrm{A}\), so it is natural for the larger number (\(n\)) to multiply \(x_1\). It is the same as a balance scale, where the balance point moves toward the heavier weight. Mixing up \(m:n\) and \(n:m\) gives a completely different point. Always write the ratio in the order "\(m\) counted from \(\mathrm{A}\), then \(n\) from there to \(\mathrm{B}\)", and you are less likely to make a mistake.

External division (the point dividing a segment externally in the ratio \(m:n\))

Figure

Standard notation (the usual math form)

\(x\) \(=\) \((\) \(-n\) \(\times\) \(x_1\) \(+\) \(m\) \(\times\) \(x_2\) \()\) \(\div\) \(m-n\)

In words (symbols replaced with words)

⑥ \(x\): x-coordinate of the point \(\mathrm{Q}\) \(=\) \((\) ① \(-n\): ratio part from the point to \(\mathrm{B}\), with its sign changed \(\times\) ② \(x_1\): x-coordinate of \(\mathrm{A}\) \(+\) ③ \(m\): ratio part from \(\mathrm{A}\) to the point \(\times\) ④ \(x_2\): x-coordinate of \(\mathrm{B}\) \()\) \(\div\) ⑤ \(m-n\): difference of the ratio parts

The formula in words

① Let \(\mathrm{Q}\) be the point dividing segment \(\mathrm{AB}\) externally in the ratio \(m:n\). Take the \(-n\): ratio part \(n\) with its sign changed

② and multiply it by the \(x_1\): x-coordinate of point \(\mathrm{A}\)

③ Take the \(m\): ratio part from \(\mathrm{A}\) to the point

④ and multiply it by the \(x_2\): x-coordinate of point \(\mathrm{B}\) (then add the two products)

⑤ Divide the sum by the \(m-n\): difference of the ratio parts

⑥ and you get the x-coordinate of the point \(\mathrm{Q}\) (when \(m\) and \(n\) are equal, the difference is \(0\), and there is no point of external division)

Quick example

The \(x\)-coordinate of the point \(\mathrm{Q}\) dividing \(\mathrm{A}(1,\ 2)\), \(\mathrm{B}(7,\ 5)\) externally in the ratio \(3:1\) is

x-coordinate of \(\mathrm{Q}\) \(=\) \((\) ratio part on the \(\mathrm{B}\) side, sign changed (−1) \(\times\) x-coordinate of \(\mathrm{A}\) (1) \(+\) ratio part on the \(\mathrm{A}\) side (3) \(\times\) x-coordinate of \(\mathrm{B}\) (7) \()\) \(\div\) difference of the ratio parts (2)

\(x = \dfrac{-1 \times 1 + 3 \times 7}{3 - 1} = \dfrac{20}{2} = 10\)

\(y = \dfrac{-1 \times 2 + 3 \times 5}{3 - 1} = \dfrac{13}{2} \quad \Rightarrow \quad \mathrm{Q}\left(10,\ \dfrac{13}{2}\right)\)

Key idea

The point of external division \(\mathrm{Q}\) lies outside segment \(\mathrm{AB}\), yet \(\mathrm{AQ} : \mathrm{QB} = m : n\). Think of internal division as dividing inside the segment, and external division as dividing on the extension of the segment. The formula looks just like the internal one, with \(n\) replaced by \(-n\). Why is the minus sign allowed? Think about direction as well as length. For internal division, going from \(\mathrm{A}\) to \(\mathrm{P}\) and going from \(\mathrm{P}\) to \(\mathrm{B}\) are in the same direction. For external division, going from \(\mathrm{A}\) to \(\mathrm{Q}\) and going from \(\mathrm{Q}\) to \(\mathrm{B}\) are always in opposite directions (\(\mathrm{Q}\) is outside the segment, so you have to turn back from \(\mathrm{Q}\) to reach \(\mathrm{B}\)). The lengths are still in the ratio \(m:n\), but with direction included, the ratio becomes \(m : \left(-n\right)\). In fact, putting \(m : \left(-n\right)\) into the internal division formula gives \(\dfrac{\left(-n\right)x_1 + m x_2}{m + \left(-n\right)} = \dfrac{-n x_1 + m x_2}{m - n}\) which is exactly the external division formula. Remember "external division is internal division with a minus sign on the second number", and you do not need to memorize two formulas. Which side the point lands on depends on which of \(m\) and \(n\) is larger. If \(m > n\), it is beyond \(\mathrm{B}\); if \(m < n\), it is before \(\mathrm{A}\) (in the direction away from \(\mathrm{B}\)). \(\mathrm{Q}\) always lands on the side of whichever point, \(\mathrm{A}\) or \(\mathrm{B}\), has the smaller ratio part. Only when \(m = n\) is there no point of external division. In the formula, the denominator \(m - n\) becomes \(0\). In the figure, there is no point on line \(\mathrm{AB}\) that is outside the segment and also equally far from \(\mathrm{A}\) and from \(\mathrm{B}\).

Midpoint (internal division in the ratio \(1:1\))

Figure

Standard notation (the usual math form)

\(x\) \(=\) \((\) \(x_1\) \(+\) \(x_2\) \()\) \(\div\) \(2\)

In words (symbols replaced with words)

③ \(x\): x-coordinate of the midpoint \(\mathrm{M}\) \(=\) \((\) ① \(x_1\): x-coordinate of \(\mathrm{A}\) \(+\) ② \(x_2\): x-coordinate of \(\mathrm{B}\) \()\) \(\div\) \(2\)

The formula in words

① Add the \(x_1\): x-coordinate of point \(\mathrm{A}\)

② and the \(x_2\): x-coordinate of point \(\mathrm{B}\) and divide by 2

③ and you get the x-coordinate of the midpoint \(\mathrm{M}\) (simply the average of the two coordinates)

Quick example

The \(x\)-coordinate of the midpoint \(\mathrm{M}\) of \(\mathrm{A}(1,\ 2)\) and \(\mathrm{B}(7,\ 5)\) is

x-coordinate of \(\mathrm{M}\) \(=\) \((\) x-coordinate of \(\mathrm{A}\) (1) \(+\) x-coordinate of \(\mathrm{B}\) (7) \()\) \(\div\) \(2\)

\(x = \dfrac{1 + 7}{2} = \dfrac{8}{2} = 4\)

\(y = \dfrac{2 + 5}{2} = \dfrac{7}{2} \quad \Rightarrow \quad \mathrm{M}\left(4,\ \dfrac{7}{2}\right)\)

Key idea

The midpoint is internal division in the ratio \(1:1\). Putting \(m = 1\) and \(n = 1\) into the internal division formula gives \(x = \dfrac{1 \times x_1 + 1 \times x_2}{1 + 1} = \dfrac{x_1 + x_2}{2}\) and the criss-cross swap disappears. Both coordinates are multiplied by the same \(1\), so it does not matter which goes with which. This is why the internal division formula suddenly feels hard after you are used to the midpoint formula: you started from this special case, where the swap is hidden. All it does is find the average of two coordinates. Think of it as the same calculation as averaging two test scores, done separately for the \(x\)-coordinates and the \(y\)-coordinates, and you will not get lost.

Centroid of a triangle

Figure

Standard notation (the usual math form)

\(x\) \(=\) \((\) \(x_1\) \(+\) \(x_2\) \(+\) \(x_3\) \()\) \(\div\) \(3\)

In words (symbols replaced with words)

④ \(x\): x-coordinate of the centroid \(\mathrm{G}\) \(=\) \((\) ① \(x_1\): x-coordinate of vertex \(\mathrm{A}\) \(+\) ② \(x_2\): x-coordinate of vertex \(\mathrm{B}\) \(+\) ③ \(x_3\): x-coordinate of vertex \(\mathrm{C}\) \()\) \(\div\) \(3\)

The formula in words

① For triangle \(\mathrm{ABC}\), add the \(x_1\): x-coordinate of vertex \(\mathrm{A}\)

② the \(x_2\): x-coordinate of vertex \(\mathrm{B}\)

③ and the \(x_3\): x-coordinate of vertex \(\mathrm{C}\) and divide by 3

④ and you get the x-coordinate of the centroid \(\mathrm{G}\) (the average of the coordinates of the three vertices)

Quick example

The \(x\)-coordinate of the centroid \(\mathrm{G}\) of the triangle with vertices \(\mathrm{A}(1,\ 2)\), \(\mathrm{B}(7,\ 5)\) and \(\mathrm{C}(4,\ 8)\) is

x-coordinate of \(\mathrm{G}\) \(=\) \((\) x-coordinate of \(\mathrm{A}\) (1) \(+\) x-coordinate of \(\mathrm{B}\) (7) \(+\) x-coordinate of \(\mathrm{C}\) (4) \()\) \(\div\) \(3\)

\(x = \dfrac{1 + 7 + 4}{3} = \dfrac{12}{3} = 4\)

\(y = \dfrac{2 + 5 + 8}{3} = \dfrac{15}{3} = 5 \quad \Rightarrow \quad \mathrm{G}(4,\ 5)\)

Key idea

The centroid \(\mathrm{G}\) is the point where the three medians (the segments joining each vertex to the midpoint of the opposite side) meet. A flat triangular plate balances level when supported at this one point, which is why it is called the center of gravity. You can check with the internal division formula why the average of the coordinates works. The \(x\)-coordinate of the midpoint \(\mathrm{M}\) of side \(\mathrm{BC}\) is \(\dfrac{x_2 + x_3}{2}\). The centroid divides the median \(\mathrm{AM}\) internally in the ratio \(2:1\), counting from the vertex, so putting \(m = 2\) and \(n = 1\) into the internal division formula gives \(x = \dfrac{1 \times x_1 + 2 \times \dfrac{x_2 + x_3}{2}}{2 + 1} = \dfrac{x_1 + x_2 + x_3}{3}\) which is the average of the three coordinates. This \(2:1\) is the most important property of the centroid: \(\mathrm{AG}\) is exactly twice as long as \(\mathrm{GM}\). The ratio is the same for every median, so remembering "the centroid divides each median \(2:1\) from the vertex" helps in geometry problems.

The point dividing segment \(\mathrm{AB}\) in the ratio \(m:n\) is \(\left(\dfrac{n x_1 + m x_2}{m+n},\ \dfrac{n y_1 + m y_2}{m+n}\right)\) for internal division, and \(\left(\dfrac{-n x_1 + m x_2}{m-n},\ \dfrac{-n y_1 + m y_2}{m-n}\right)\), with \(n\) replaced by \(-n\), for external division. The coordinate of point \(\mathrm{A}\) is multiplied by \(n\), the number for the side far from \(\mathrm{A}\), so that the larger \(n\) is, the closer the point gets to \(\mathrm{A}\) (the dividing point moves toward the end with the smaller ratio part). The midpoint is internal division in the ratio \(1:1\), and the centroid of a triangle is the average of the coordinates of its three vertices. They are all connected by the same idea: a weighted average.

Symbols and terms

Symbols

\(\mathrm{A},\ \mathrm{B},\ \mathrm{C}\)	A, B, C	Names for points. In geometry, points are named with capital letters, usually A, B, C in order. On this page, the ends of the segment are \(\mathrm{A}\) and \(\mathrm{B}\), and the third vertex of the triangle is \(\mathrm{C}\).
\(x_1,\ y_1\)	x sub one, y sub one	The coordinates of point \(\mathrm{A}\). The small number at the lower right (the subscript) is a label telling which point it is, not a number used in the calculation. In the same way, \(x_2,\ y_2\) are the coordinates of point \(\mathrm{B}\), and \(x_3,\ y_3\) those of point \(\mathrm{C}\).
\(m:n\)	m to n	The ratio that divides the segment. \(m\) is the share for the part from \(\mathrm{A}\) to the dividing point, and \(n\) the share for the part from the dividing point to \(\mathrm{B}\). \(m\) and \(n\) are simply letters often used for ratios. Swapping the order gives a different point.
\(\mathrm{P}\)	P	A name often given to the point of internal division, from the first letter of "point".
\(\mathrm{Q}\)	Q	A name often given to the point of external division. By custom, it is the letter after \(\mathrm{P}\).
\(\mathrm{M}\)	M	A name often given to the midpoint, from the first letter of "midpoint".
\(\mathrm{G}\)	G	A name often given to the centroid. It is said to come from the "G" in "center of gravity".
\(\mathrm{AB}\)	segment AB	The line segment joining point \(\mathrm{A}\) and point \(\mathrm{B}\): the part of a straight line between two fixed ends. On this page, its length is also written \(\mathrm{AB}\).
\(\mathrm{AP} : \mathrm{PB}\)	AP to PB	The ratio of the length of segment \(\mathrm{AP}\) to the length of segment \(\mathrm{PB}\). The point \(\mathrm{P}\) with \(\mathrm{AP} : \mathrm{PB} = m : n\) is the point dividing segment \(\mathrm{AB}\) internally in the ratio \(m:n\).

Terms

line segment	The part of a straight line between two points. A part with two ends is a line segment, one that goes on forever in one direction is a ray, and one that goes on forever in both directions is a line.
internal division	Dividing a segment at a point inside it in a given ratio. The point \(\mathrm{P}\) dividing segment \(\mathrm{AB}\) internally in the ratio \(m:n\) is the point on segment \(\mathrm{AB}\) with \(\mathrm{AP} : \mathrm{PB} = m : n\). In US geometry this is called partitioning a directed line segment.
external division	Dividing a segment at a point on its extension, outside the segment, so that \(\mathrm{AQ} : \mathrm{QB} = m : n\). The point lands on the side of the end with the smaller ratio part (beyond \(\mathrm{B}\) if \(m > n\), before \(\mathrm{A}\) if \(m < n\)). There is no such point when \(m\) and \(n\) are equal.
dividing point	A general name for a point that divides a segment in a given ratio, covering both internal and external division.
midpoint	The point in the middle of a segment, equally far from both ends. It is internal division in the ratio \(1:1\).
centroid	The one point where the three medians of a triangle meet. A triangular plate balances level when supported there (its center of gravity), and its coordinates are the average of the coordinates of the three vertices.
median	The segment joining a vertex of a triangle to the midpoint of the opposite side. The three medians always meet at one point (the centroid), which divides each median internally in the ratio \(2:1\) from the vertex.
ratio	A way of comparing two amounts, written side by side as in "\(2:1\)". \(2:1\) and \(4:2\) are the same ratio. On this page, it compares the lengths of the two parts of a divided segment.
coordinates	Numbers that give the position of a point: one number on the number line, two \((x,\ y)\) in the plane, and three \((x,\ y,\ z)\) in space.
number line	A straight line with a scale marked on it, so that each number matches exactly one point. It is the simplest kind of coordinate, where one number gives a position.
coordinate plane	A plane with a horizontal \(x\)-axis and a vertical \(y\)-axis, so that any point on it can be given by two numbers \((x,\ y)\).
weighted average	An average where each number is given a "weight". The coordinate of internal division \(\dfrac{n x_1 + m x_2}{m+n}\) is exactly the weighted average of \(x_1\) with weight \(n\) and \(x_2\) with weight \(m\). An ordinary average is the special case where all the weights are equal.
subscript	A small number written at the lower right of a letter, as in \(x_1\) and \(x_2\). It is a label that tells apart quantities of the same kind; it is neither a factor nor an exponent.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the fastest way forward.

The number line and coordinates (Grades 4–6)	Knowing that a number line is a straight line with a scale where numbers match points, and being able to find where negative numbers are Being able to read the point \((3,\ -2)\) on the coordinate plane as "3 to the right and 2 down" Knowing that when the coordinates of point \(\mathrm{A}\) are written with letters and subscripts, as in \((x_1,\ y_1)\), the subscript \(1\) is a label for "the first point", not a number to multiply by
Ratios and proportions (Grades 6–7)	Knowing that a ratio such as \(2:1\) splits a whole into 3 equal parts, shared out as 2 and 1 Knowing that \(2:1\) and \(4:2\) are the same ratio (multiplying or dividing both parts by the same number does not change the ratio) Knowing that when a segment is divided in the ratio \(m:n\), the part on the \(\mathrm{A}\) side is \(\dfrac{m}{m+n}\) of the whole
Operations with positive and negative numbers (Grade 7)	Being able to get the sign right when multiplying and dividing negative numbers, as in \(-3 \times 4 = -12\) and \(\left(-9\right) \div \left(-2\right) = \dfrac{9}{2}\) Being able to calculate with a letter that has a minus sign, such as the \(-n\) in the external division formula
Working with fractions (Grades 4–6)	Being able to simplify, as in \(\dfrac{15}{3} = 5\), and write answers as fractions in lowest terms Being comfortable leaving a fraction such as \(\dfrac{9}{2}\) as the answer (no need to change it to a decimal)
Algebraic expressions and substitution (Grades 6–8)	Being able to substitute numbers into an expression such as \(\dfrac{n x_1 + m x_2}{m + n}\) and calculate Being able to expand an expression such as \(x_1 + \dfrac{m}{m+n}\left(x_2 - x_1\right)\) and combine like terms
Averages (Grade 6)	Knowing that an average is found by adding everything up and dividing by how many there are (the midpoint and the centroid are both exactly this kind of average) Knowing that to combine the averages of groups of different sizes, you cannot simply add them and divide

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the point of internal division

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
Ratio part m (from A to the point)	2
Ratio part n (from the point to B)	1
x-coordinate of P (internal)	=(B6B1+B5B3)/(B5+B6)
y-coordinate of P (internal)	=(B6B2+B5B4)/(B5+B6)

Table to find the point of external division

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
Ratio part m (from A to the point)	3
Ratio part n (from the point to B)	1
x-coordinate of Q (external)	=(-B6B1+B5B3)/(B5-B6)
y-coordinate of Q (external)	=(-B6B2+B5B4)/(B5-B6)

Table to find the midpoint

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
x-coordinate of midpoint M	=(B1+B3)/2
y-coordinate of midpoint M	=(B2+B4)/2

Table to find the centroid of a triangle

Vertex A x-coordinate x1	1
Vertex A y-coordinate y1	2
Vertex B x-coordinate x2	7
Vertex B y-coordinate y2	5
Vertex C x-coordinate x3	4
Vertex C y-coordinate y3	8
x-coordinate of centroid G	=(B1+B3+B5)/3
y-coordinate of centroid G	=(B2+B4+B6)/3

After pasting, the upper rows (the coordinates and the ratio) are your inputs, and the lower rows are calculated automatically. "*" is multiplication and "/" is division.
The first table divides A(1, 2), B(7, 5) internally in the ratio 2:1, giving P(5, 4). The formula for the x-coordinate, "=(B6*B1+B5*B3)/(B5+B6)", looks mismatched at first because it multiplies B1 (the x-coordinate of A) by B6 (the ratio part n), and B3 (the x-coordinate of B) by B5 (the ratio part m). This is the criss-cross, and swapping them changes the answer.
The second table divides the same two points externally in the ratio 3:1, giving Q(10, 6.5). The only differences from the first table are the minus sign in front of B6 and the subtraction instead of addition in the denominator. If you enter the same value for m and n, the denominator becomes 0 and an error (#DIV/0!) appears; this tells you there is no point of external division.
The third table gives the midpoint M(4, 3.5). The fourth table gives the centroid G(4, 5) of the triangle with vertices A(1, 2), B(7, 5) and C(4, 8).

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the point of internal division

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
Ratio part m (from A to the point)	2
Ratio part n (from the point to B)	1
x-coordinate of P (internal)	=(B6B1+B5B3)/(B5+B6)
y-coordinate of P (internal)	=(B6B2+B5B4)/(B5+B6)

Table to find the point of external division

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
Ratio part m (from A to the point)	3
Ratio part n (from the point to B)	1
x-coordinate of Q (external)	=(-B6B1+B5B3)/(B5-B6)
y-coordinate of Q (external)	=(-B6B2+B5B4)/(B5-B6)

Table to find the midpoint

Point A x-coordinate x1	1
Point A y-coordinate y1	2
Point B x-coordinate x2	7
Point B y-coordinate y2	5
x-coordinate of midpoint M	=(B1+B3)/2
y-coordinate of midpoint M	=(B2+B4)/2

Table to find the centroid of a triangle

Vertex A x-coordinate x1	1
Vertex A y-coordinate y1	2
Vertex B x-coordinate x2	7
Vertex B y-coordinate y2	5
Vertex C x-coordinate x3	4
Vertex C y-coordinate y3	8
x-coordinate of centroid G	=(B1+B3+B5)/3
y-coordinate of centroid G	=(B2+B4+B6)/3

The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the coordinates and the ratio with your own numbers.

How to calculate it in Python

from fractions import Fraction

# Coordinates of the points (a fraction such as 3/4 can be written Fraction(3, 4))
point_a = (Fraction(1), Fraction(2))
point_b = (Fraction(7), Fraction(5))
point_c = (Fraction(4), Fraction(8))   # third vertex, used for the centroid

def internal_point(a, b, m, n):
    # Internal division: multiply by the number for the far side (n for point A, m for point B)
    return tuple((n * p + m * q) / (m + n) for p, q in zip(a, b))

def external_point(a, b, m, n):
    # External division: internal division with n replaced by -n (cannot be calculated when m equals n)
    return tuple((-n * p + m * q) / (m - n) for p, q in zip(a, b))

def midpoint(a, b):
    # Midpoint: the average of the two coordinates (internal division 1:1)
    return tuple((p + q) / 2 for p, q in zip(a, b))

def centroid(a, b, c):
    # Centroid of a triangle: the average of the coordinates of the three vertices
    return tuple((p + q + r) / 3 for p, q, r in zip(a, b, c))

def show(point):
    # Show the fractions in an easy-to-read form such as "(5, 4)" or "(10, 13/2)"
    return "(" + ", ".join(str(v) for v in point) + ")"

print("Internal 2:1 :", show(internal_point(point_a, point_b, Fraction(2), Fraction(1))))
print("External 3:1 :", show(external_point(point_a, point_b, Fraction(3), Fraction(1))))
print("Midpoint     :", show(midpoint(point_a, point_b)))
print("Centroid     :", show(centroid(point_a, point_b, point_c)))

The fractions module in the standard library lets you calculate with exact fractions, with no decimal rounding errors. This example uses A(1, 2), B(7, 5) and C(4, 8). When you run it, it shows the point of internal division (5, 4), the point of external division (10, 13/2), the midpoint (4, 7/2) and the centroid (4, 5). Because it uses zip, it also works in space (3D) as is if you give each point three coordinates.

How to write it in LaTeX and other math languages (copy and paste)

Internal division (the point dividing a segment internally in the ratio \(m:n\))

P = ((n·x₁ + m·x₂) ÷ (m + n),  (n·y₁ + m·y₂) ÷ (m + n))

P\left(\dfrac{n x_1 + m x_2}{m + n},\ \dfrac{n y_1 + m y_2}{m + n}\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>x</mi>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <mi>n</mi><msub><mi>x</mi><mn>1</mn></msub>
        <mo>+</mo>
        <mi>m</mi><msub><mi>x</mi><mn>2</mn></msub>
      </mrow>
      <mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow>
    </mfrac>
  </mrow>
</math>

x = (n*x_1 + m*x_2)/(m + n)

{(n*x1 + m*x2)/(m + n), (n*y1 + m*y2)/(m + n)}

P := [(n*x1 + m*x2)/(m + n), (n*y1 + m*y2)/(m + n)];

P = [(n*x1 + m*x2)/(m + n), (n*y1 + m*y2)/(m + n)];

x = (n x_1 + m x_2)/(m + n)

External division (the point dividing a segment externally in the ratio \(m:n\))

Q = ((−n·x₁ + m·x₂) ÷ (m − n),  (−n·y₁ + m·y₂) ÷ (m − n))

Q\left(\dfrac{-n x_1 + m x_2}{m - n},\ \dfrac{-n y_1 + m y_2}{m - n}\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>x</mi>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <mo>&#x2212;</mo><mi>n</mi><msub><mi>x</mi><mn>1</mn></msub>
        <mo>+</mo>
        <mi>m</mi><msub><mi>x</mi><mn>2</mn></msub>
      </mrow>
      <mrow><mi>m</mi><mo>&#x2212;</mo><mi>n</mi></mrow>
    </mfrac>
  </mrow>
</math>

x = (-n*x_1 + m*x_2)/(m - n)

{(-n*x1 + m*x2)/(m - n), (-n*y1 + m*y2)/(m - n)}

Q := [(-n*x1 + m*x2)/(m - n), (-n*y1 + m*y2)/(m - n)];

Q = [(-n*x1 + m*x2)/(m - n), (-n*y1 + m*y2)/(m - n)];

x = (-n x_1 + m x_2)/(m - n)

Midpoint (internal division in the ratio \(1:1\))

M = ((x₁ + x₂) ÷ 2,  (y₁ + y₂) ÷ 2)

M\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>x</mi>
    <mo>=</mo>
    <mfrac>
      <mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow>
      <mn>2</mn>
    </mfrac>
  </mrow>
</math>

x = (x_1 + x_2)/2

{(x1 + x2)/2, (y1 + y2)/2}

M := [(x1 + x2)/2, (y1 + y2)/2];

M = [(x1 + x2)/2, (y1 + y2)/2];

x = (x_1 + x_2)/2

Centroid of a triangle

G = ((x₁ + x₂ + x₃) ÷ 3,  (y₁ + y₂ + y₃) ÷ 3)

G\left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>x</mi>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <msub><mi>x</mi><mn>1</mn></msub><mo>+</mo>
        <msub><mi>x</mi><mn>2</mn></msub><mo>+</mo>
        <msub><mi>x</mi><mn>3</mn></msub>
      </mrow>
      <mn>3</mn>
    </mfrac>
  </mrow>
</math>

x = (x_1 + x_2 + x_3)/3

{(x1 + x2 + x3)/3, (y1 + y2 + y3)/3}

G := [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3];

G = [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3];

x = (x_1 + x_2 + x_3)/3

How to have ChatGPT do the calculation

You are a math calculation assistant for coordinate geometry. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

For the two points A(1, 2) and B(7, 5) on the coordinate plane, find:
1. The coordinates of the point P dividing segment AB internally in the ratio 2:1
2. The coordinates of the point Q dividing segment AB externally in the ratio 3:1
3. The coordinates of the midpoint M of segment AB
4. The coordinates of the centroid G of triangle ABC, with the added point C(4, 8)

In Python, use the fractions module from the standard library to calculate exactly, and give each answer both as a fraction in lowest terms and as a decimal.
Also explain, by following the algebra step by step, why the coordinate of point A is multiplied by n in the internal division formula (why the formula is "criss-crossed").

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula