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
