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Sum Reciprocal Lengths (Posted on 2007-11-17) Difficulty: 2 of 5
Each of the sides PQ, QR, RS, ST, TU, UV and VP of a regular heptagon PQRSTUV have length 2. The respective lengths of two of the diagonals PR and PS of the heptagon are denoted by X and Y.

Determine 1/X + 1/Y.

See The Solution Submitted by K Sengupta    
Rating: 2.5000 (2 votes)

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Solution | Comment 2 of 6 |

Consider the quadrilateral defined by points P, Q, R, and T.  The sides are PQ=2, QR=2, RT=X, TP=Y and the diagonals are PR=X and QT=Y.

Then apply Ptolemy's Theorem
http://mathworld.wolfram.com/PtolemysTheorem.html

2*X + 2*Y = X*Y

1/Y + 1/X = 1/2


  Posted by Brian Smith on 2007-11-17 20:58:44
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