Kazakhstan National Olympiad 2014 P4 D2 11 grade

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#1 h Mar 22, 2014, 11:58 AM • 2 Y

Y by Adventure10, Mango247

Given a scalene triangle $ABC$ . Incircle of $\triangle{ABC{}}$ touches the sides $AB$ and $BC$ at points $C_1$ and $A_1$ respectively, and excircle of $\triangle{ABC}$ (on side $AC$ ) touches $AB$ and $BC$ at points $C_2$ and $A_2$ respectively. $BN$ is bisector of $\angle{ABC}$ ( $N$ lies on $BC$ ). Lines $A_1C_1$ and $A_2C_2$ intersects the line $AC$ at points $K_1$ and $K_2$ respectively. Let circumcircles of $\triangle{BK_1N}$ and $\triangle{BK_2N}$ intersect circumcircle of a $\triangle{ABC}$ at points $P_1$ and $P_2$ respectively. Prove that $AP_1$ = $CP_2$

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#2 h Mar 22, 2014, 4:20 PM • 2 Y

Y by Adventure10, Mango247

let $M$ is midpoint of smaller arc $AC$ of $(ABC)$ . Then $M,P_1,K_1$ and $M,P_2,K_2$ are collinear. Since $K_1A=K_2C$ we get that $MK_1=MK_2$ and $\angle MK_1A=\angle MK_2C$ . Hence $\angle P_1BN=\angle P_2BN$ and $\angle ABP_1=\angle CBP_2$ $\Rightarrow$ $AP_1=CP_2$ .

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