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## Nice problem about Mixtilinear incircle

Let (I) and (O) be the incircle and circumcircle of triangle ABC, respectively, P be an arbitrary point on (O). Construct two tangents from P to (I), they cut BC at E and F. Prove that (PEF) passes through the tangency of A-Mixtilinear incircle of triangle ABC and (O).

## Two lines and one circle are concurrent

Given triangle $ABC$ and its circumcenter $O$. Let $P$ be an arbitrary point on the plane, $A_1B_1C_1$ be the pedal triangle of $P$ wrt $\Delta ABC. (A_1B_1C_1)$ intersects 9-point circle of triangle $ABC$ at $X_1$ and $X_2$. By Fontene’s theorem, we know that $X_1$ is the Anti-Steiner point of $OP$ wrt median triangle of triangle $ABC$. Let $P'$ be the isogonal conjugate of $P$ wrt $\Delta ABC, M$ be the midpoint of $AP', M_a$ be the midpoint of $BC, H$ be the projection of $A$ onto $BC. (MX_1H)\cap BC=\{H,E\}$, $(MX_1H)\cap (A_1B_1C_1)=\{F\}$. Prove that the intersection of $EF$ and $M_aX_2$ lies on $(A_1B_1C_1)$.

Given triangle $ABC$ and its circumcircle $(O)$. Let $P$ be an arbitrary point in the plane. $AP, BP, CP$ cuts $(O)$ again at $A_1, B_1, C_1$, respectively. Let $A_2, B_2, C_2$ be the reflections of $P$ wrt $\Delta ABC$. Prove that 4 circles $(O), (PA_1A_2), (PB_1B_2), (PC_1C_2)$ are concurrent.