MathDB
Areas, quadrilaterals and lots of points

Source: JBMO Shortlist 2002

November 12, 2008
geometryrectanglegeometry proposed

Problem Statement

Let ABCD ABCD be a convex quadrilateral with AB\equal{}AD and BC\equal{}CD. On the sides AB,BC,CD,DA AB,BC,CD,DA we consider points K,L,L1,K1 K,L,L_1,K_1 such that quadrilateral KLL1K1 KLL_1K_1 is rectangle. Then consider rectangles MNPQ MNPQ inscribed in the triangle BLK BLK, where M∈KB,N∈BL,P,Q∈LK M\in KB,N\in BL,P,Q\in LK and M1N1P1Q1 M_1N_1P_1Q_1 inscribed in triangle DK1L1 DK_1L_1 where P1 P_1 and Q1 Q_1 are situated on the L1K1 L_1K_1, M M on the DK1 DK_1 and N1 N_1 on the DL1 DL_1. Let S,S1,S2,S3 S,S_1,S_2,S_3 be the areas of the ABCD,KLL1K1,MNPQ,M1N1P1Q1 ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1 respectively. Find the maximum possible value of the expression: \frac{S_1\plus{}S_2\plus{}S_3}{S}
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