AoPS - Problem

Source: Problem 4 Geometry

Tags: geometry, incenter, power of a point, radical axis, geometry unsolved

shoki

13.09.2009 15:29

4-Point $P$ is taken on the segment $BC$ of the scalene triangle $ABC$ such that $AP \neq AB,AP \neq AC$ . $l_1,l_2$ are the incenters of triangles $ABP,ACP$ respectively. circles $W_1,W_2$ are drawn centered at $l_1,l_2$ and with radius equal to $l_1P,l_2P$ ,respectively. $W_1,W_2$ intersects at $P$ and $Q$ . $W_1$ intersects $AB$ and $BC$ at $Y_1( \mbox{the intersection closer to B})$ and $X_1$ ,respectively. $W_2$ intersects $AC$ and $BC$ at $Y_2(\mbox{the intersection closer to C})$ and $X_2$ ,respectively.PROVE THE CONCURRENCY OF $PQ,X_1Y_1,X_2Y_2$ .

shoki

18.09.2009 20:49

nobody?

yetti

20.09.2009 22:42

Let $R \equiv X_1Y_2 \cap X_2Y_2.$ $BX_1 = BY_1, CX_2 = CY_2$ and $AY_1 = AP = AY_2$ $\Longrightarrow$ $\angle Y_2Y_1X_1 = 180^\circ - (\angle AY_1Y_2 + \angle X_1Y_1B) =$ $= \frac {_1}{^2}\angle A + \frac{_1}{^2}\angle B = 90^\circ - \frac {_1}{^2} \angle C = \angle Y_2X_2C$ $\Longrightarrow$ $X_1X_2Y_2Y_1$ is cyclic $\Longrightarrow$ $\overline {RX_1} \cdot \overline{RY_1} = \overline {RX_2} \cdot \overline{RY_2}$ $\Longrightarrow$ $R$ is on radical axis $PQ$ of the circles $\mathcal W_1, \mathcal W_2.$