End Point Calculator

What this calculator does

So, you know one end of a line segment and its midpoint, and you want the other end. This tool finds it. Enter the starting point and the midpoint, and it returns the coordinates of the missing endpoint.

Four inputs, the x and y of the start and of the midpoint, and the far end comes back.

How to use it

1. Enter the starting point coordinates, the end you know.
2. Enter the midpoint coordinates.
3. Press Calculate.

Working backwards from the middle

Normally you find a midpoint from two ends. This tool solves the opposite puzzle: given one end and the middle, where is the other end? The key insight is that the midpoint is always the same distance from both ends, sitting exactly between them. So whatever step takes you from the starting point to the midpoint, taking that same step again carries you on to the far endpoint. The midpoint is halfway, so the other end is one more half-step beyond it.

The formula, and why it works

In coordinates, the rule is short: the missing endpoint's x is two times the midpoint's x minus the starting x, and the same for the y. This comes straight from reversing the midpoint calculation. The midpoint's x is the average of the two ends, meaning it is the sum of the two x values divided by two. Rearranging that to solve for the unknown end gives twice the midpoint minus the known end. The tool applies this to both the x and the y, doubling the midpoint and subtracting the start.

A worked example

Suppose the starting point is 2 comma 3 and the midpoint is 5 comma 7. For the x of the far end, take two times 5, which is 10, and subtract 2 to get 8. For the y, take two times 7, which is 14, and subtract 3 to get 11. So the other endpoint is 8 comma 11. Checking, the midpoint of 2 comma 3 and 8 comma 11 is indeed 5 comma 7.

The reverse of the midpoint

This tool is the mirror image of the midpoint calculator. That one takes two endpoints and averages them to find the middle; this one takes the middle and one end and works backwards to the other end. They undo each other: feed the result of one into the other and you return to where you started. If you have two ends and want the middle, reach for the midpoint calculator. If you have the middle and one end, this is the tool.

Where this comes up

Finding a missing endpoint is useful whenever a centre is known but a boundary is not. If you know where the middle of a shape sits and where one corner is, you can locate the opposite corner. In navigation, if a halfway point and a starting position are known, this gives the destination. It is the natural tool for any situation framed around a centre and a single known end, where the second end is what you are after.

Questions people ask

What does this tool find?

The other end of a segment, given one end and the midpoint.

How does it work it out?

It doubles the midpoint and subtracts the starting point, for both the x and the y coordinate.

Why double the midpoint and subtract?

Because it reverses the averaging that defines the midpoint. Solving the midpoint formula for the unknown end gives twice the midpoint minus the known end.

How can I check the answer?

Find the midpoint of your starting point and the result. It should match the midpoint you entered.

What if I have both ends and want the middle?

Then use the midpoint calculator, which averages the two endpoints.

References

On finding an endpoint. The missing end of a segment is found by reversing the midpoint formula, doubling the midpoint and subtracting the known end.

1. Eric W. Weisstein, "Midpoint," from MathWorld, a Wolfram resource, on the midpoint formula that this calculation reverses.
2. Eric W. Weisstein, "Line," from MathWorld, a Wolfram resource, on lines and segments in the coordinate plane.