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

Solution

The proportion blue : green is 5 : 3.

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