MathDB
existence of a point P [Iran Second Round 1993]

Source:

November 25, 2010
geometrygeometry proposed

Problem Statement

Let ABCABC be an acute triangle with sides and area equal to a,b,ca, b, c and SS respectively. [color=#FF0000]Prove or disprove that a necessary and sufficient condition for existence of a point PP inside the triangle ABCABC such that the distance between PP and the vertices of ABCABC be equal to x,yx, y and zz respectively is that there be a triangle with sides a,y,za, y, z and area S1S_1, a triangle with sides b,z,xb, z, x and area S2S_2 and a triangle with sides c,x,yc, x, y and area S3S_3 where S1+S2+S3=S.S_1 + S_2 + S_3 = S.
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