A trapezium containing three right angles is split in two. What is the proportion of the areas yellow : blue?

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## Solution

The ratio yellow : blue is (1+√5)/2.

From the right angles at B and C we infer that DC and AB are parallel. Therefore we can equate angles ABD and BDC. Setting the two equal sides to 1, we can easily find the bottom side and diagonal as shown.

Now from the Pythagorean theorem we get 1+1/cos^{2}(α)=1/sin^{2}(α). Using simple trigonometry, we find one positive solution sin(α)=1/φ (the Golden Ratio). Since the yellow and blue triangles have the same height, their area proportion is simply their base proportion, which is φ.

## Poem

The three rights

The right to know

The right to participate

The right to refuse

Or the three rights

As figure of geometry

They were in love

Touching eachother

One time in angle

One time in corner

They sometimes cut their way

But don’t like to be secant lines

To play