Monday, March 2, 2009

Problem 261: Regular Pentagon inscribed in a circle, sum of distances

Proposed Problem
Problem 261: Regular Pentagon inscribed in a circle, sum of distances.

See complete Problem 261 at:
gogeometry.com/problem/p261_regular_pentagon_inscribed_circle_distance.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

1 comment:

  1. ABCDE is a regular pentagon;hence all its sides are equal. Let it's side = x. Hence BD = 2a*sin(54). Let 2sin(54) = s. Thus, BD =AC=AD= x*s.
    Apply Ptolemy's Theorem to BFCD,AFCD and FCDE to obtain: b*x+sc*x=d*x or b+sc=d --(1)Similarly, a+sc=sd --(2)and c + e = sd--(3) . Add (1) and (3) to get b+c+e+sc = sd + d or b+c+e = sd-sc+d = a + d by equation (2). QED
    Vihaan: vihaanup@gmail.com

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