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2011 OIM Problems/Problem 3

Revision as of 14:54, 14 December 2023 by Tomasdiaz (talk | contribs) (Created page with "== Problem == Let <math>ABC</math> be a triangle and let <math>X, Y, Z</math> be the points of tangency of its circumference inscribed with sides <math>BC</math>, <math>CA</ma...")
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Problem

Let $ABC$ be a triangle and let $X, Y, Z$ be the points of tangency of its circumference inscribed with sides $BC$, $CA$, $AB$ respectively. Suppose $C_1, C_2, C_3$ are circles with chords $YZ$, $ZX$, $XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose $C_1$ cuts chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ cuts the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ cuts the chords $YZ$ and $ZX$ at $K$ and $N$ respectively. Show that $I, J, K, L, M, N$ are on a circle.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions

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