The tip of the iceberg

A triangle with a 45° angle is placed inside a square as shown. What’s the area proportion of parts A and B?

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A and B have equal area , so their proportion is 1.

Draw the vertical diagonal and then consider triangles AGB and CFB. Their top angle is seen to be equal and they also both have a 45° angle. Hence all their angles are equal and the triangles are similar.

Now from proportions in the square, their longest sides have proportion 1: √2, and therefore also BG : FB = 1 : √2. The same reasoning can be applied to the other side of A and the total triangle, and since the area in both triangles is half the product of the sides times sin(45) it follows that A is half the total triangle. B is the other half.

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