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Hexagonal Dilemma (Posted on 2004-06-02) Difficulty: 4 of 5
A hexagon with sides of length 2, 7, 2, 11, 7, 11 is inscribed in a circle. Find the radius of the circle.

As suggested, *if* it matters, you may assume that the sides listed are given in order

No Solution Yet Submitted by SilverKnight    
Rating: 4.0000 (5 votes)

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Solving the arccos equation. | Comment 13 of 18 |
(In reply to No Subject by Gamer)

I also got the equation: arccos(1-(2/x)/2)+ arccos(1-(7/x)/2)+ arccos(1-(11/x)/2) = pi

To solve it, first take sin()'s of each side and use trig identities to get the equivalent equation: (2/x)*(sqrt(1-4/(4x^2)))*(1-49/(2x^2))*)))*(1-121/(2x^2)) + (7/x)*(sqrt(1-49/(4x^2)))*(1-4/(2x^2))*)))*(1-121/(2x^2)) + (11/x)*(sqrt(1-121/(4x^2)))*(1-4/(2x^2))*)))*(1-49/(2x^2)) - (154/x^3)*(sqrt(1-4/(4x^2)))*(sqrt(1-49/(4x^2)))*(sqrt(1-121/(4x^2))) = 0

A little bit of algebra reduces the equation to: (8x^4 - 680x^2 + 11858)*sqrt(4x^2-4) + (28x^4 - 1750x^2 + 3388)*sqrt(4x^2-49) + (44x^4 - 1166x^2 + 2156)*sqrt(4x^2-121) - 154*sqrt(4x^2-4)*(sqrt(4x^2-49)*sqrt(4x^2-121).

To evaluate this monster equation, I used the polynomial abilities of UBASIC to simplify the polynomials generated by isolating the roots and squaring.  The equation left over after eliminating common factors is: 3840x^8 - 310364x^6 + 5202948x^4 + 39184761x^2 - 35153041 = 0.  7 is indeed a root of the equation.

  Posted by Brian Smith on 2004-06-04 08:46:20
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