A triangle having one side tangent to the circle is split into two triangles. What is the proportion blue : purple?

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## Solution

The proportion blue : purple is (1+√5)/2.

First, this solution uses the Tangent-chord theorem to fix the purple top angle to 36°. Using the fact that purple is isosceles and the interior angle sum, the other angles can be found.

We see that the blue triangle is also isosceles and is part of a regular pentagon, of which it is known that the ratio of the diagonal to the side is the golden ratio φ.

Since the combined blue-purple triangle is also isosceles, we find that 1+x=φ. Since they have the same height, the proportion of blue : purple is that of their bases 1 : x, which evaluates to φ.

## Poem

The blue face mask

Hiding our emotions

Just in name of prevention

We breathe like a doctor or a nurse

And have all the time a mask in the purse

Not easy and certainly not funny

But it should keep us healthy

We must wear it in a shop

And this situation is not yet gone to stop