A right triangle is divided in three right triangles. What is the area proportion A : B : C?

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## Solution

The areas have proportion 3 : 2 : 1.

This solution starts by setting the equal line segments to length 1. From similarity of triangle A to the whole triangle we get 1/x=x/2, so x=β2.

Therefore similar triangles A and C have side proportion 1 : β2 : β3. The hypothenuse of C is 1, therefore we get for the areas A=3C.

Since the combined triangle B+C has the same area as A (same base and height), it follows that B=2C.