AUSTRIAN-POLISH 2006 Problem 6

Let $D$ be an interior point of the triangle $ABC$ . $CD$ and $AB$ intersect at $D_{c}$ , $BD$ and $AC$ intersect at $D_{b}$ , $AD$ and $BC$ intersect at $D_{a}$ . Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$ , so that $(AD_{c}D) = (KH_{m}H)$ $(BD_{c}D) = (LH_{m}H)$ $(BD_{a}D) = (LH_{k}H)$ $(CD_{a}D) = (MH_{k}H)$ $(CD_{b}D) = (MH_{l}H)$ $(AD_{b}D) = (KH_{l}H)$

where $(PQR)$ denotes the area of the triangle $PQR$