Proposed Problem

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.

## Monday, March 2, 2009

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

Labels:
circumcircle,
inscribed,
pentagon,
Ptolemy,
regular polygon

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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.

ReplyDeleteApply 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

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