How to use the section point calculator
Choose the relationship you know
Use Internal point when both endpoints A and B are known and you need a point P between them. Enter the positive ratio as AP:PB = m:n. The default 1:1 ratio is the midpoint. A ratio 2:1 places P two-thirds of the way from A toward B because AP is twice PB.
Use Other endpoint when you know endpoint A and midpoint P. In this mode the second coordinate pair is the midpoint, not point B, and no ratio fields are needed. The calculator reflects A across P to recover B. The two modes belong to the same segment relationship, but the labels change so an endpoint is not accidentally entered as a midpoint.
Internal division formula
For A(x₁, y₁), B(x₂, y₂), and AP:PB = m:n, the internal point is P((n·x₁ + m·x₂)/(m+n), (n·y₁ + m·y₂)/(m+n)). This is a weighted average. The ratio part m multiplies B because a larger AP moves P closer to B; n multiplies A for the same reason from the other side.
Both ratio parts must be greater than zero. That condition keeps P strictly between distinct endpoints. Multiplying m and n by the same positive factor does not move P: 2:4 and 1:2 describe the same position.
Midpoint and endpoint formula
A midpoint is the average P = ((xA+xB)/2, (yA+yB)/2). Solving separately for the unknown coordinates gives xB = 2xP − xA and yB = 2yP − yA. The result panel substitutes the entered values so you can verify the reflection across P.
The diagram labels A, P, and B and scales all three to the available area. It shows ordering and collinearity rather than a physical unit. The result text can be copied; changing the mode or any input immediately invalidates the old result.
Worked examples
Internal point in ratio 2:1
Let A(2, 4), B(8, 10), and AP:PB = 2:1. P = ((1·2 + 2·8)/3, (1·4 + 2·10)/3) = (6, 8). P is closer to B, matching the longer AP part.
Midpoint as ratio 1:1
For A(−4, 2) and B(6, 8), P = ((−4+6)/2, (2+8)/2) = (1, 5). Equal ratio parts produce the ordinary midpoint.
Recover the other endpoint
If A(−3, 4) and midpoint P(2, 1) are known, B = (2·2−(−3), 2·1−4) = (7, −2). Averaging A and B returns P(2, 1).
Uses, limits, and output
Appropriate uses
Use the calculator for analytic geometry, placing a point at a known fraction along a segment, checking midpoint work, simple interpolation, or recovering a missing endpoint. Coordinates may be negative or decimal values as long as they share the same coordinate system.
Supported boundaries
Ratio mode supports positive internal division only. A zero or negative part is rejected because it would describe an endpoint, external division, or a different convention. The tool does not calculate external section points, triangle centres, multi-segment paths, or general geometry constructions.
Distinct endpoints are required. If A and B are identical, a direction and meaningful AP:PB split are not defined; if A equals midpoint P in endpoint mode, the recovered B would also be the same point. Extreme results outside the finite range are rejected.
Frequently asked questions
Which segment does m describe?
m is the AP part in AP:PB = m:n. It multiplies B in the weighted-average formula.
Why does 2:1 move P toward B?
AP is twice PB, so the trip from A to P is longer and P must lie nearer B.
Can I enter 0:1 or a negative ratio?
No. This page is limited to positive internal division. Zero and negative parts describe boundaries or external division.
How do I find a midpoint?
Keep the default ratio 1:1 in Internal point mode.
How can I verify a recovered endpoint?
Average A and the calculated B coordinate by coordinate. The result should equal the entered midpoint P.