Let $\displaystyle P$ be a point inside triangle $\displaystyle \Delta ABC$, and lines through $\displaystyle \{A,P\},\{B,P\},\{C,P\}$

intersect the sides of $\displaystyle \Delta ABC$ at $\displaystyle A_1, B_1, C_1$. Let $\displaystyle A_2, B_2, C_2$ be the symmetric points of

$\displaystyle P$ on segments $\displaystyle \overline{AA}_1, \overline{BB}_1, \overline{CC}_1 $ with respect to their middle points $\displaystyle A_m,B_m, C_m$

Show that $\displaystyle |\Delta A_1 B_1 C_1| = |\Delta A_2 B_2 C_2|$