How this instrument works
A midpoint is the average of two points: M = (A + B) / 2, one equation applied separately to the x-coordinates and to the y-coordinates. That averaging step is reversible without any loss of information — multiply the midpoint by two and subtract the point you already know, and the equation resolves cleanly to the point you don't: B = 2M − A. Nothing about the segment's length or tilt matters here; the formula only tracks position.
Geometrically, the missing endpoint is the known endpoint spun 180 degrees about the midpoint — a point reflection, sometimes called central symmetry. Walk from the known point to the midpoint, then keep walking the same distance in the same direction, and you land exactly on the other endpoint. Doubling the midpoint's coordinates and subtracting the known point is that walk written as algebra instead of steps.
A useful edge case: if the known endpoint already sits on the midpoint, the two coordinates are identical and the formula hands that same point straight back — a segment of zero length, which is a valid answer rather than a fault. The formula is also indifferent to sign; endpoints with negative coordinates, or a midpoint parked at the origin, fall out of the same two subtractions with no special-casing anywhere.
- Enter the segment's midpoint coordinates into Midpoint: x and Midpoint: y.
- Enter the coordinates of the endpoint you already know into Known endpoint: x and Known endpoint: y.
- Read Other endpoint: x and Other endpoint: y — the point sitting directly across the midpoint from the one you entered.
- Check the answer by averaging Known endpoint: x with Other endpoint: x; it should land back on Midpoint: x, and the same check applies to the y-pair.
Worked example — midpoint (4, 4), known endpoint (1, 2)
Suppose a fence line's midpoint is surveyed at (4, 4) and one marked corner post stands at (1, 2), with the matching post buried or unreadable. Doubling the midpoint gives (8, 8); subtracting the known post's coordinates, (8 − 1, 8 − 2), leaves the missing post at exactly (7, 6).
The check confirms it: average the two endpoints, (1 + 7) / 2 = 4 and (2 + 6) / 2 = 4, and the midpoint (4, 4) reappears exactly, with no rounding anywhere in the arithmetic. The distance from the known post to the midpoint, and from the midpoint onward to (7, 6), is identical in both direction and length — the point-reflection idea made concrete with numbers a tape measure could check.
Questions
What formula finds a missing endpoint from a midpoint?
B = 2M − A, applied separately to each coordinate: x2 = 2·mx − x1 and y2 = 2·my − y1. It comes directly from solving the midpoint formula M = (A + B) / 2 for B, which is why doubling the midpoint and subtracting the known point always works, regardless of the segment's length or direction.
How is this different from just calculating a midpoint?
The midpoint formula moves forward, from two endpoints to their average; this calculator runs the same equation backward, from an average and one endpoint to the other endpoint. Both use identical algebra, but which pieces you already have determines which unknown you can solve for.
Why does doubling the midpoint work to find the other point?
Because a midpoint is defined as an average, multiplying it by two undoes the averaging and returns the sum of both endpoints: 2·mx = x1 + x2. Subtracting the endpoint you already know leaves only the endpoint you don't, isolated by ordinary algebra with no approximation involved.
What if the known endpoint and the midpoint are the same point?
Then the segment has zero length, and the calculator correctly returns the other endpoint as that same point too, since 2M − A equals A whenever A already equals M. This is not an error state; a single point is a valid degenerate segment whose two endpoints and midpoint all coincide.
Can this method be extended to three dimensions?
Yes — the same doubling-and-subtracting step works on a z-coordinate exactly as it does on x and y, since the midpoint formula averages each axis independently. This sheet handles the two-dimensional case; a third axis would simply add z2 = 2·mz − z1 alongside the results shown here.
Does it matter which endpoint I label as known?
No — swapping which point is called known and which is other changes nothing about the geometry, only which coordinates you type in and which the sheet solves for. The midpoint sits at the same halfway position no matter which endpoint gets entered first.