KöMaL 2024 C1804

In triangle $ABC$ let $D$, $E$ and $F$ denote the midpoints of the sides $BC$, $CA$ and $AB$, respectively. Let $K_a$, $K_b$ and $K_c$ denote the centers of the incircles of triangles $AFE$, $BDF$ and $CED$, respectively. Prove that the sum of the areas of quadrilaterals $K_aFDE$, $K_bDEF$ and $K_cEFD$ equals the area of triangle $ABC$.