Let-ABC-be-an-acute-angled-triangle-with-AC-BC-and-let-O-be-the-circumcenter-and-F-be-the-foot-of-altitude-through-C-Further-let-X-and-Y-be-the-feet-of-perpendiculars-dropped-from-A-and-B-respecti

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Question Number 19104 by Tinkutara last updated on 04/Aug/17

Let ABC be an acute-angled triangle with AC ≠ BC and let O be the circumcenter and F be the foot of altitude through C. Further, let X and Y be the feet of perpendiculars dropped from A and B respectively to (the extension of) CO. The line FO intersects the circumcircle of ΔFXY, second time at P. Prove that OP < OF.

$$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{FO}\:\mathrm{intersects} \\ $$$$\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\Delta\mathrm{FXY},\:\mathrm{second}\:\mathrm{time} \\ $$$$\mathrm{at}\:\mathrm{P}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{OP}\:<\:\mathrm{OF}. \\ $$

Commented by Tinkutara last updated on 05/Aug/17