AoPS - Problem

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Tags: geometry, incenter, geometry unsolved

momo1729

28.04.2011 14:29

Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.

$\angle AEB=\angle ACB,$ $\angle ADB=\angle AFB$ and $CD=EF$ implies that $\triangle AEF$ and $\triangle ACD$ are congruent by ASA $\Longrightarrow$ $AE=AC$ and $AD=AF.$ Consequently, $\triangle AEC$ and $\triangle AFD$ are similar A-isosceles triangles, which yields $\angle ABC=\angle ABF,$ i.e. $BA$ bisects $\angle FBC$ internally, thus $CM,FN$ and $BA$ concur at the incenter $I$ of $\triangle BCF.$ Hence, $\overline{IC} \cdot \overline{IM}=\overline{IB} \cdot \overline{IA}=\overline{IF} \cdot \overline{IN}$ $\Longrightarrow$ $C,F,M,N$ are concyclic.