# 2012 JBMO Problems/Problem 2

## Section 2

Let the circles and intersect at two points and , and let be a common tangent of and that touches and at and respectively. If and , evaluate the angle .

## Solution

Let and be the centers of circles and respectively. Also let be the intersection of and line .

Note that is perpendicular to since is a tangent of . In order for to be perpendicular to , must be the point diametrically opposite . Note that is a right angle since it inscribes a diameter. By AA similarity, . This gives that .

By Power of a Point on point with respect to circle , we have that . Using Power of a Point on point with respect to circle gives that . Therefore and . Since , . We now see that is a triangle. Since it is similar to , $\angle PMB \cong \boxed {\angle NMB \cong 45 \degree \cong \frac{\pi}{4}}$ (Error compiling LaTeX. ! Undefined control sequence.).