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Constructing triangles holding many similarities

Source: Baltic Way 2011

November 6, 2011
geometry proposedgeometry

Problem Statement

Let EE be an interior point of the convex quadrilateral ABCDABCD. Construct triangles △ABF,△BCG,△CDH\triangle ABF,\triangle BCG,\triangle CDH and △DAI\triangle DAI on the outside of the quadrilateral such that the similarities △ABF∼△DCE,△BCG∼△ADE,△CDH∼△BAE\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE and △DAI∼△CBE \triangle DAI\sim\triangle CBE hold. Let P,Q,RP,Q,R and SS be the projections of EE on the lines AB,BC,CDAB,BC,CD and DADA, respectively. Prove that if the quadrilateral PQRSPQRS is cyclic, then EF⋅CD=EG⋅DA=EH⋅AB=EI⋅BC.EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.
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