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maximum product of distances from sides of a regular pentagon

Source: Norwegian Mathematical Olympiad 2015 - Abel Competition p3

September 3, 2019
RegularpentagondistancemaximumProductgeometry

Problem Statement

The five sides of a regular pentagon are extended to lines ℓ1,ℓ2,ℓ3,ℓ4\ell_1, \ell_2, \ell_3, \ell_4, and ℓ5\ell_5. Denote by did_i the distance from a point PP to ℓi\ell_i. For which point(s) in the interior of the pentagon is the product d1d2d3d4d5d_1d_2d_3d_4d_5 maximal?
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