A regular hexagon and two equilateral triangles share a vertex. What is the area proportion blue : yellow?
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Solution
The proportion blue : yellow is 3.
First, draw the diagonal EB. Since ∠FEB is 60°, it is easy to show that ∠FEI=∠BEK. Since also ∠IFE=∠KBE=60°, it follows from equal angles that △FIE and △BKE are similar.
Now, because |BE|=2|FE| it follows that |KE|=2|IE|. The yellow equilateral triangle thus is 1/4 of the yellow-blue one. The proportion follows.
Poem
The pyramid is swinging
And the old stones are dancing
In a sand beach surrounding
Saqqara with Gizeh is meeting
And skylights are shining
The old stones are laughing
Cause in the dark they are feeling
Happy and in love together choreographing
A great love story beginning
