How to use the division point calculator
This calculator finds the point that divides the segment between A(x₁, y₁) and B(x₂, y₂) in the ratio m : n. The internal division point sits inside the segment, while the external division point sits outside it on the extended line. Enter the six values and both points are computed at once, along with the midpoint and the length of AB.
The internal point is ((n·x₁ + m·x₂)/(m+n), (n·y₁ + m·y₂)/(m+n)). People often mix up which letter multiplies which coordinate; the easy way to remember is that m, the share nearer B, multiplies x₂. The external point is the same formula with n replaced by −n, giving ((m·x₂ − n·x₁)/(m−n), (m·y₂ − n·y₁)/(m−n)).
Because the external denominator is m − n, the external point is undefined when m and n are equal. In that case the row is hidden and the reason is explained. A note also says whether the external point falls beyond B (when m > n) or beyond A (when m < n). If all six inputs are whole numbers the coordinates are shown as reduced fractions too, so you see 11/2 rather than only 5.5. The ratio values cannot be zero or negative, and three-dimensional coordinates are not supported.
Frequently Asked Questions
The internal point is ((n·x₁ + m·x₂)/(m+n), (n·y₁ + m·y₂)/(m+n)); the external point flips the sign of n and becomes ((m·x₂ − n·x₁)/(m−n), (m·y₂ − n·y₁)/(m−n)).
The external denominator is m − n, so m = n would divide by zero. In that case the tool hides the external row and says it is undefined.
The 1 : 1 internal division point is the midpoint. Enter 1 and 1 for the ratio and the internal point matches the midpoint exactly.