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2008 IMO Problems/Problem 1

Revision as of 11:03, 14 July 2011 by Motiejus (talk | contribs) (Created page with "== Problem 1 == An acute-angled triangle <math>ABC</math> has orthocentre <math>H</math>. The circle passing through <math>H</math> with centre the midpoint of <math>BC</math> in...")
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Problem 1

An acute-angled triangle $ABC$ has orthocentre $H$. The circle passing through $H$ with centre the midpoint of $BC$ intersects the line $BC$ at $A_1$ and $A_2$. Similarly, the circle passing through $H$ with centre the midpoint of $CA$ intersects the line $CA$ at $B_1$ and $B_2$, and the circle passing through $H$ with centre the midpoint of $AB$ intersects the line $AB$ at $C_1$ and $C_2$. Show that $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ lie on a circle.

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