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Pentagon Fraction Minimizing (Posted on 2015-03-16) Difficulty: 3 of 5
ABCDE is a regular pentagon. Determine the smallest value of the expression

(|PA|+|PB|)/(|PC|+|PD|+|PE|)

where P is an arbitrary point lying in the plane of the pentagon ABCDE.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 1 of 2
Moving point P around in Geometer's Sketchpad, I find the minimum for the fraction occurs whenever P lies on the small arc AB of the circle circumscribing the pentagon. In GSP, the value of that fraction is then given as 0.23607.

Doing some trigonometry, and assuming WLOG that the point lies at the middle of arc AB, we need find only one side of the symmetric set, so that we calculate |PA|/(|PC| + |PD|/2), so that both the numerator and the denominator have been divided by 2.

Doing the trig, assuming a unit circumscribing circle, it comes out to

  2*sin(18°) / (2*sin(54°) + 1) ~= 0.23606797749979.

  Posted by Charlie on 2015-03-16 13:16:19
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