Wherever the Point Lands — Opposite Triangles Match
Pick any point inside a rectangle and join it to the four corners. The two pairs of opposite triangles always have equal area — one auxiliary line proves it.
Problem
From an arbitrary interior point P, draw segments to all four corners, forming four triangles. Compare (top + bottom) with (left + right).
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
From any interior point P to the four corners — four triangles.
Why this approach works
P's coordinates are unknown, but one auxiliary line settles it. Draw a vertical through P: the top and bottom triangles share base W, and their heights sum to H. So top + bottom = ½·W·H = half the rectangle. A horizontal line shows left + right = half too. Both halves — so they're equal, wherever P sits.
Proof
A vertical line shows the top/bottom heights sum to H; a horizontal line shows the left/right widths sum to W. Each pair is half the rectangle, so they are equal regardless of P.
Try it yourself
Slide P into a corner: one triangle nearly vanishes while its partner grows, yet top + bottom stays exactly half.
Related mathematics
Beyond MathVoyage
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