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Existence of triangle

Source: APMC 2006, Problem 6

September 9, 2006
geometrygeometry proposed

Problem Statement

Let DD be an interior point of the triangle ABCABC. CDCD and ABAB intersect at DcD_{c}, BDBD and ACAC intersect at DbD_{b}, ADAD and BCBC intersect at DaD_{a}. Prove that there exists a triangle KLMKLM with orthocenter HH and the feet of altitudes Hk∈LM,Hl∈KM,Hm∈KLH_{k}\in LM, H_{l}\in KM, H_{m}\in KL, so that (ADcD)=(KHmH)(AD_{c}D) = (KH_{m}H) (BDcD)=(LHmH)(BD_{c}D) = (LH_{m}H) (BDaD)=(LHkH)(BD_{a}D) = (LH_{k}H) (CDaD)=(MHkH)(CD_{a}D) = (MH_{k}H) (CDbD)=(MHlH)(CD_{b}D) = (MH_{l}H) (ADbD)=(KHlH)(AD_{b}D) = (KH_{l}H) where (PQR)(PQR) denotes the area of the triangle PQRPQR
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