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area of triangle conctructed by 3 perpendiculars on sides ABC

Source: Korean Mathematical Olympiad 1995, Final Round P4 FKMO

July 21, 2018
geometryperpendicular linesarea of a triangle

Problem Statement

Let OO and RR be the circumcenter and circumradius of a triangle ABCABC, and let PP be any point in the plane of the triangle. The perpendiculars PA1,PB1,PC1PA_1,PB_1,PC_1 are drawn from PP on BC,CA,ABBC,CA,AB. Express SA1B1C1/SABCS_{A_1B_1C_1}/S_{ABC} in terms of RR and d=OPd = OP, where SXYZS_{XYZ} is the area of △XYZ\triangle XYZ.
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