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.
Do not write an equation yet. Shake the figure first.
Drag the point somewhere unreasonable
The four triangles deform wildly, yet both opposite pairs refuse to leave one half.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
From any interior point P to the four corners — four triangles.
Return to the exact question
From an arbitrary interior point P, draw segments to all four corners, forming four triangles. Compare (top + bottom) with (left + right).
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.
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