Question 3
Let triangle $ABC$ be inscribed in circle $(O)$, and let $D$ lie on the arc $BC$ not containing $A$. Suppose $CD$ meets $AB$ at $E$, and $BD$ meets $AC$ at $F$. Let $(K)$ be the circle inside triangle $EBD$, tangent to $EB,ED$ and to $(O)$. Let $(L)$ be the circle inside triangle $FCD$, tangent to $FC,FD$ and to $(O)$.
(a) Let $M$ be the contact of $(K)$ with $BE$ and $N$ the contact of $(L)$ with $CF$. Prove that the circle with diameter $MN$ always passes through a fixed point as $D$ moves.
(b) The line through $M$ parallel to $CE$ meets $AC$ at $P$; the line through $N$ parallel to $BF$ meets $AB$ at $Q$. Prove that the circumcircles of $AMP$ and $ANQ$ are both tangent to a fixed circle as $D$ moves.
Source wording note: the source accidentally calls $(L)$ a “center”; its subsequent tangencies and the solution clearly use it as a circle. The full-arc formulation has a configuration issue, explicitly discussed in the solution.
(a) Let $M$ be the contact of $(K)$ with $BE$ and $N$ the contact of $(L)$ with $CF$. Prove that the circle with diameter $MN$ always passes through a fixed point as $D$ moves.
(b) The line through $M$ parallel to $CE$ meets $AC$ at $P$; the line through $N$ parallel to $BF$ meets $AB$ at $Q$. Prove that the circumcircles of $AMP$ and $ANQ$ are both tangent to a fixed circle as $D$ moves.
Source wording note: the source accidentally calls $(L)$ a “center”; its subsequent tangencies and the solution clearly use it as a circle. The full-arc formulation has a configuration issue, explicitly discussed in the solution.