ILL 1974 Problem 4

Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$ , touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively. Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$ . Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$ , where $I$ is the incentre of $ABC$ .