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Estonian Math Competitions

Geometry Difficulty 4.9 AIME Prove it Estonia

On the sides $AB$ , $BC$ and $CA$ of triangle $ABC$ , points $L$ , $M$ and $N$ are chosen, respectively, such that the lines $CL$ , $AM$ and $BN$ intersect at a common point $O$ inside the triangle and the quadrilaterals $ALON$ , $BMOL$ and $CNOM$ have incircles. Prove that
$\frac{1}{AL \cdot BM} + \frac{1}{BM \cdot CN} + \frac{1}{CN \cdot AL} = \frac{1}{AN \cdot BL} + \frac{1}{BL \cdot CM} + \frac{1}{CM \cdot AN}.$

Solution

$ALON$ is a circumscribed quadrilateral, hence $AL + ON = AN + OL$ . Similarly $BM + OL = BL + OM$ and $CN + OM = CM + ON$ . By adding the equations we obtain $AL + BM + CN = AN + BL + CM$ . Lines $CL$ , $AM$ and $BN$ intersect in one point, we get from Ceva's theorem $AL \cdot BM \cdot CN = AN \cdot BL \cdot CM$ . By dividing the left hand sides of the last two equations with the right hand sides we obtain the required equation.

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