Three rectangles and four triangles. Given that the red triangles have equal area, prove that the blue rectangles are similar.

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## Solution

Naming the sides of the rectangles as shown. Now the lower red triangle has area k*l*a*b*sin(180-α)/2, whereas the central yellow triangle has area a*b*sin(α)/2. It is not hard then to show that red is k*l*yellow. Likewise the other red triangles are l*m*yellow and k*m*yellow.

Since they are given to be equal, we have k*l=l*m=k*m, which can be simplified to k=l=m. This implies similarity of the three rectangles.

## Poem

The crazy diamond

He’s like glass or made of crystal

He’s unique, a jewel

Cut of the mountains

Out of the stones

So brightly clear

So beautiful, so dear

But the nicest diamond is life

probably

Using the day and time surely

The life is

As precious as gold

A gift to hold !