The right angle

Two triangles inside a quarter circle. What is their area proportion?

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The proportion blue : green is 5 : 3.

Solution by David Andriana.

First, extend the quarter circle to a semicircle and use Thales to show that the line FE goes through D. Consider triangle DGC. Comparing angles, it is similar to BGF. Their sides have proportion 1 : 2 : √5. It follows that BGF fits five times into DGC.

Since it is also rather straightforward that CEG fits three times into DGC, the ratio 5 : 3 follows.

Visual solution

Proof without words by David Andriana.

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