How do I find the other endpoint from one end and the midpoint?

B = 2M − A. Double the midpoint and subtract the end you know. The reasoning is worth holding on to rather than the formula: the midpoint is the average, so (A + B) ÷ 2 = M, and undoing that means multiplying by two before subtracting. Doing it in the other order — subtracting first — is the standard mistake and it produces a point near the middle rather than at the far end.

Why is M − A wrong?

Because it computes the gap from A to the midpoint rather than the position of the far end. It happens to be the right vector and the wrong point — you would need to add it to the midpoint, not use it on its own. On integer coordinates across a small grid the two give different answers 99.7% of the time. The cases where they agree are exactly those with the midpoint at the origin, since setting 2M − A equal to M − A forces M to be zero, and that coincidence is what makes the mistake survive the occasional check.

Why is my midpoint a fraction?

Because the formula halves, and half of an odd number is not whole. A midpoint has whole coordinates only when the two x values agree in parity and the two y values do as well — both even or both odd on each axis. Each of those is a coin flip, so roughly a quarter of integer endpoint pairs give a whole midpoint and the other three quarters have a half somewhere. A half is the exact answer, not an approximation of one.

Does the order of the two points matter?

No, because addition does not care about order. Swapping the ends gives the same sum and therefore the same average. This is a genuine difference from formulas built on subtraction, where the order flips the sign of the result even when it does not change the final answer — here there is no sign to flip and nothing to be careful about.

What is the section formula?

The general version: the point dividing a segment m parts from the first end and n parts from the second is (nA + mB) ÷ (m+n). Watch which coefficient goes where — the far end is multiplied by the first part of the ratio, so a 3:1 point is three quarters of the way along and nearer the second end. Setting m equal to n makes the coefficients match and the formula becomes the plain average, so the midpoint is the 1:1 member of this family rather than a separate rule.

Is the midpoint always on the segment?

Yes, and so is any point from a positive ratio — the construction is a weighted average of the two ends, and a weighted average with positive weights always lands between them. That is worth knowing as a check: if a section point comes out beyond one of the ends, the arithmetic went wrong somewhere, because no positive ratio can produce that.

How does the midpoint work in three dimensions?

Average the z values as well: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). Because the coordinates never interact, adding a dimension adds a line of arithmetic and changes nothing else. The same is true of four dimensions and beyond, where the picture stops working long before the formula does.

What if both points are the same?

The midpoint is that point. The segment has no length, so its middle is where it already is. This is a correct answer rather than an error, and the same is true in the inverse direction: if the end you know is also the midpoint, the other end must be that point too, which the formula gives you without a special case.

Is the midpoint the same as the centre of several points?

Not quite. Averaging two points gives their midpoint; averaging three or more gives their centroid, which is a different construction with different properties. For a triangle the centroid is where the medians meet and it sits two thirds of the way along each of them, not at the middle of any side. The midpoint formula is the two-point case of that averaging, but the geometry stops being about halves as soon as a third point joins.

Midpoint Formula Calculator

The midpoint formula averages each coordinate on its own: M = ((x₁+x₂)/2, (y₁+y₂)/2). The two axes never interact, which is why it extends to three dimensions without changing and why an error in one coordinate leaves the other perfectly correct — easy to miss when checking by eye. Finding a midpoi...

Given one end and the midpoint, the other end is 2M − A, not M − A. Halving got you halfway, so reaching the far end means going that distance again — double the midpoint and take the known end away. Subtracting straight off looks reasonable and lands you back near the middle, and it gives a different answer on 99.7% of cases. The two agree only when the midpoint happens to sit at the origin. This page does that direction as well as the ordinary one, and the general point that divides a segment in any ratio, of which the midpoint is the 1:1 case.

THE TWO ENDS

Whole numbers, decimals and fractions such as 3/4 all work.

X₁

Y₁

X₂

Y₂

THE MIDPOINT

(2, 3)

Each coordinate averaged on its own, and the result is exactly 2 from each end along x.

THE ANSWER PUT BACK

x: (0 + 4) ÷ 2 = 2 ✓

y: (0 + 6) ÷ 2 = 3 ✓

x: 2 from each end ✓

y: 3 from each end ✓

FROM EACH END

(2, 3)

the same gap on both sides

WHOLE COORDINATES

yes

both ends agreed in parity

QUARTER POINT

(1, 3/2)

midway from the first end to the middle

THREE-QUARTER POINT

(3, 9/2)

midway from the middle to the far end

WHAT IS HAPPENING

1Average each coordinate on its own

x: (0 + 4) ÷ 2 = 2 · y: (0 + 6) ÷ 2 = 3

The two axes do not interact. Adding the pair and halving is done once for x and once for y, and nothing in one calculation affects the other.

2Check it lands in the middle

x: 2 either side · y: 3 either side

The gap from each end to the midpoint has to be the same, and that is a real check rather than a restatement — it recomputes from the answer rather than repeating the step that produced it.

The axes never interact. Averaging the two x values and averaging the two y values are separate calculations that happen to be written on one line, and neither affects the other. That is why the formula extends to three dimensions without changing — a third average joins the other two — and why a mistake in one coordinate leaves the other perfectly correct, which makes it easy to miss when checking by eye.

Exact fractions throughout · the answer is averaged back to check it returns the midpoint

Created with❤️byeaglecalculator.com

HOW TO USE

1
Pick the direction you need. The first tab finds the midpoint from two ends; the second finds the missing end from one end and the midpoint; the third divides the segment in any ratio.
2
Enter the coordinates. Whole numbers, decimals and fractions all work, and the arithmetic stays exact — you will see 5/2 rather than 2.5.
3
In the inverse mode, read the warning panel. It shows what M − A would have given, so the wrong answer and the right one can be compared directly rather than described.
4
Check the substitution panel. It averages the two ends back together and confirms the result is the midpoint, which is a real check rather than a restatement of the step above it.

WORKED EXAMPLE

Take (0, 0) and (4, 6). Average each coordinate on its own: x gives (0 + 4) ÷ 2 = 2, and y gives (0 + 6) ÷ 2 = 3. The midpoint is (2, 3). Check it by measuring: 2 is two from each end along x, and 3 is three from each end along y. Now (1, 2) and (4, 7), where the numbers are less obliging. x gives (1 + 4) ÷ 2 = 5/2 and y gives (2 + 7) ÷ 2 = 9/2, so the midpoint is (5/2, 9/2). Both halves. Nothing has gone wrong — the formula divides by two every time, and a whole answer needs both ends to agree in parity. That happens for one coordinate about half the time and for both about a quarter of the time, so three midpoints in four have a half in them. Now the direction that causes trouble. One end is (1, 2), the midpoint is (4, 7), and the other end is missing. The midpoint is the average, so (1 + ?) ÷ 2 = 4. Undo the halving first: 1 + ? = 8. Then undo the addition: ? = 7. The same for y gives 12. The other end is (7, 12), and the general rule is B = 2M − A. Compare that with the shortcut. Taking M − A gives (4 − 1, 7 − 2) = (3, 5), which is a perfectly reasonable-looking point and completely wrong. Check it the honest way: average (1, 2) with (3, 5) and you get (2, 7/2), which is not the midpoint you were given. Average (1, 2) with (7, 12) and you get (4, 7), which is. Finally, dividing in a ratio. From (0, 0) to (6, 9), the point one third of the way along is the 1:2 point — one part from the first end, two parts to the second. It is (2·0 + 1·6) ÷ 3 = 2 for x and (2·0 + 1·9) ÷ 3 = 3 for y, so (2, 3). Change the ratio to 2:1 and the point moves to (4, 6), two thirds along. Change it to 1:1 and the coefficients match: (1·0 + 1·6) ÷ 2 = 3 and (1·0 + 1·9) ÷ 2 = 9/2, giving (3, 9/2) — which is exactly what the midpoint formula gives for those two ends.

REFERENCE RULES

Rule	What it says	Why
The midpoint formula	((x₁+x₂)/2, (y₁+y₂)/2)	Average each coordinate on its own. The axes never interact.
Order does not matter	addition is commutative	Swapping the ends gives the same midpoint, unlike a subtraction.
The missing endpoint	B = 2M − A	Double the midpoint, subtract the known end. Not M − A.
Why doubling	halving got you halfway	Reaching the far end means going the same distance again.
The common mistake	M − A lands near the middle	It differs from the right answer on 99.7% of cases.
When they coincide	only if M is the origin	2M − A = M − A forces M = 0. A neat exact fact, and rarely useful.
Whole midpoints are rare	a quarter of integer pairs	Both x values must agree in parity, and both y values too.
Halves, not decimals	5/2 rather than 2.5	The formula always halves, so exact fractions cost nothing.
The section formula	(nA + mB) ÷ (m+n)	The point m:n along the segment. The midpoint is the 1:1 case.
Which coefficient goes where	the far end takes m	A 3:1 point sits three quarters of the way, nearer the second end.
Both ends the same	a segment of no length	The midpoint is that point. A valid answer, not an error.
In three dimensions	average z as well	The pattern extends unchanged to any number of coordinates.
The midpoint is on the segment	always between the ends	True of any positive ratio, not only 1:1.
Not the same as the centre	of three or more points	Averaging n points gives their centroid, a different construction.

FREQUENTLY ASKED QUESTIONS

M = ((x₁+x₂)/2, (y₁+y₂)/2) — average the two x values and, separately, average the two y values. The word separately is the important one: the x calculation and the y calculation share nothing, which is why the same pattern extends to three dimensions by adding a third average and to any number of coordinates after that.

·Average the x values, then average the y values
·The two calculations are entirely independent
·A third coordinate just adds a third average
·The pattern continues in higher dimensions

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Last updated: August 6, 2026 · Exact fractions throughout, never floating point · Given one end and the midpoint, the other end is 2M − A.