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ratio of triangles concerning touchpoints of incircle and perpendiculars

Source: IGO 2014 Junior 2

July 22, 2018
geometryareasincircleperpendicularratio

Problem Statement

The inscribed circle of △ABC\triangle ABC touches BC,ACBC, AC and ABAB at D,ED,E and FF respectively. Denote the perpendicular foots from F,EF, E to BCBC by K,LK, L respectively. Let the second intersection of these perpendiculars with the incircle be M,NM, N respectively. Show that S△BMDS△CND=DKDL\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}
by Mahdi Etesami Fard
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