If you want a math quiz : A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. Find the radius of the circle.
http://www.qbyte.org/puzzles/p063s.html
Trigonometric Solution
We can drop a perpendicular from the center of the circle to each of the chords, bisecting the isosceles triangles, as shown.
We have a + b + c = /2, and c + d = /2.
Hence a + b = d < /2.
We have a + b + c = /2, and c + d = /2.
Hence a + b = d < /2.
We also have
Taking the cosine of both sides of a + b = d, and using trigonometric identities cos(x + y) = cosx cosy − sinx siny, and sin2x + cos2x = 1, we get
Adding 7/2r2 to both sides of the equation, squaring, and multiplying by 2r3, we obtain
2r3 − 87r − 77 = 0
This easily factorizes, giving (r − 7)(2r2 + 14r + 11) = 0.
The quadratic factor has negative real roots. Hence r = 7 is the only positive real root.
The quadratic factor has negative real roots. Hence r = 7 is the only positive real root.
Therefore the radius of the circumscribing circle of the original hexagon is 7 units.
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