IRL_ABooklet

Solution 1. Draw the tangent to ${\Omega }_1$ at $A$ and let $H$ be its second point of intersection with ${\Omega }_2$. First suppose $P$ is not inside circle ${\Omega }_2$.



By the Alternate Segment Theorem, $\angle HAE=\angle PBA$. From the cyclic quadrilateral $ABFE$ we see that $\angle PBA=\angle AEF$, hence $\angle HAE=\angle AEF$. We also have $\angle FAH=\angle FEH$ (standing on the same arc of ${\Omega }_2$). Therefore, $$\angle FAE=\angle FAH+\angle HAE=\angle FEH+\angle AEF=\angle AEH.$$ This implies that the segment $EF$ has the same length as $AH$. If $P$ is inside circle ${\Omega }_2$, we can proceed in a similar way.



Looking at triangle $PBE$ we find that $\angle FBE=\angle BPA-\angle BEA$. The Alternate Segment Theorem shows that the angle between $AB$ and the tangent $AH$ is equal to $\angle BPA$. From cyclic quadrilateral $ABEH$ we see that the same angle between $AB$ and the tangent $AH$ is equal to $\angle BEH$. Hence, $$\angle FBE=\angle BPA-\angle BEA=\angle BEH-\angle BEA=\angle AEH.$$ Therefore, for each $P$ on ${\Omega }_1$, segments $EF$ and $AH$ have the same length.

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Alternative solution.

Solution 2. The length of the chord EF of circle ${\Omega }_2$ is determined by angle $\angle FAE$. First suppose $P$ is not inside circle ${\Omega }_2$.



In this case $\angle FAE$ is an external angle of triangle $PFA$, hence is equal to $\angle APF+\angle AFP$. These two angles are the same for each point $P$ on ${\Omega }_1$, because $\angle APF$ stands on the arc $AB$ of circle ${\Omega }_1$, and $\angle AFP$ stands on the arc $AB$ of circle ${\Omega }_2$.



If $P$ is on the arc of ${\Omega }_1$ that is inside the circle ${\Omega }_2$, the angle $\angle FAE$ is an internal angle of triangle $PFA$ and so is equal to $180^{\circ }-(\angle APF+\angle AFP)$. This angle subtends the same chord as the angle $\angle APF+\angle AFP$, which does not change when $P$ varies, and which has the same value as in the other case. Hence, $\angle FAE$ is the same for each point $P$ on ${\Omega }_1$.