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SAMO Problem 4: Relationship between area of triangles

Source: South African Mathematics Olympiad 2018, Problem 4

July 27, 2018
geometrycircumcirclearea

Problem Statement

Let ABCABC be a triangle with circumradius RR, and let ℓA,ℓB,ℓC\ell_A, \ell_B, \ell_C be the altitudes through A,B,CA, B, C respectively. The altitudes meet at HH. Let PP be an arbitrary point in the same plane as ABCABC. The feet of the perpendicular lines through PP onto ℓA,ℓB,ℓC\ell_A, \ell_B, \ell_C are D,E,FD, E, F respectively. Prove that the areas of DEFDEF and ABCABC satisfy the following equation: area⁡(DEF)=PH24R2⋅area⁡(ABC). \operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).
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