Problem

Source: Iran 3rd round 2012-Geometry exam-P1

Tags: ratio, geometry, circumcircle, trigonometry, parallelogram, geometry proposed



Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. Proposed by Mehdi E'tesami Fard