Two equilateral triangles and a quarter circle that goes through both apexes. If the smaller triangle has area 1, what is the total shaded area?
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Solution
The shaded area is 5.
First extend the quarter to a full circle. Then reflect the large triangle in the diameter as shown. The angles on the left side of B add up to 180. Therefore, DE is a straight line.
From Thales we know that angle DEC is right. Now consider triangle BEC. It is a 30-60-90 triangle, and it is well known that the short side is exactly half the hypothenuse. So |CB|=2|EB|. The large triangle is therefore four times larger than the small one.
Visual solution
Poem
Gizeh and it’s pyramids
Keops, Kephren, Mykerinos
So beautiful
And aesthetic
So powerfull
And dynamic
A treasure for generations
A gift for civilisations
With 1000 angles of beauty
Climbing the steps, heavenly
To the egyptian dynasty
A glorious hymn to trinity
Gizeh is eternity
