VN-TST 2007 Problem 2

Let $ABC$ be an acute triangle with incricle $(I)$ . $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$ , similary we have $B_{1},C_{1}$ . a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$ . b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$ .

Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$ .