# Difference between revisions of "2009 IMO Problems/Problem 2"

Line 17: | Line 17: | ||

&=\text{Pow}_{\omega}(P)\\ | &=\text{Pow}_{\omega}(P)\\ | ||

&=R^2-PO^2. | &=R^2-PO^2. | ||

− | \end{align*}</cmath> It follows that <math> | + | \end{align*}</cmath> It follows that <math>OP=OQ.</math> <math>\blacksquare</math> |

## Revision as of 12:04, 1 April 2016

## Problem

Let be a triangle with circumcentre . The points and are interior points of the sides and respectively. Let and be the midpoints of the segments and , respectively, and let be the circle passing through and . Suppose that the line is tangent to the circle . Prove that .

*Author: Sergei Berlov, Russia*

## Solution

By parallel lines and the tangency condition, Similarly, so AA similarity implies Let denote the circumcircle of and its circumradius. As both and are inside

It follows that