Question 126
Let $P$ be a point in the interior of the triangle $ABC$, and let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ respectively at the points $D, E, F$. Let the circles on diameters $BC$ and $AD$ meet at points $a$ and $a'$; the circles on diameters $CA$ and $BE$ meet at points $b$ and $b'$; and the circles on diameters $AB$ and $CF$ meet at points $c$ and $c'$. Show that the points $a, a', b, b', c, c'$ lie on a circle.