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point in triangle, distance to sides and vertices (Bulgaria 1962 P4)

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June 24, 2021
geometryTriangles

Problem Statement

There are given a triangle and some internal point PP. x,y,zx,y,z are distances from PP to the vertices A,BA,B and CC. p,q,rp,q,r are distances from PP to the sides BC,CA,ABBC,CA,AB respectively. Prove that: xyz≥(q+r)(r+p)(p+q).xyz\ge(q+r)(r+p)(p+q).
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