The figure shows a cyclic quadrilateral ABCD, AC and BD meet at
E. O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6} are circles with diameters AB, BC, CD,
AD, AC, and BD, respectively. P_{12}, P_{14}, P_{15}, P_{16}, P_{23}, P_{25}, P_{26},
P_{34}, P_{35}, P_{36}, P_{45}, P_{46}, P_{56}, P_{56*} are the points of
intersection of circles O_{1} and O_{2}, O_{1} and O_{4}, O_{1} and O_{5}, O_{1} and
O_{6}, O_{2} and O_{3}, O_{2} and O_{5}, O_{2} and O_{6}, O_{3} and O_{4}, O_{3} and O_{5}, O_{3}
and O_{6}, O_{4} and O_{5}, O_{4} and O_{6}, and O_{5} and O_{6}, respectively.
FG is the common chord of circles O_{2} and O_{4}. Prove
that (1) P_{16}, P_{12}, and P_{26} are collinear (lie on a line L_{126}),
similarly P_{15}, P_{14}, and P_{45} are collinear (lie on a line L_{145}),
P_{25}, P_{23}, and P_{35} are collinear (lie on a line L_{235}), P_{46}, P_{34},
and P_{36} are collinear (lie on a line L_{346}); (2) P_{56}, E,
and P_{56*} are
collinear (line on a line L_{56}); (3) Lines L_{126}, L_{145}, L_{235},
L_{346}, L_{56}, and FG are concurrent at P.
