MathDB
circle with tangent lines

Source: IMO Shortlist 1994, G5

April 15, 2004
geometrycircumcirclereflectiontrigonometryrotationIMO Shortlist

Problem Statement

A circle C C with center O. O. and a line L L which does not touch circle C. C. OQ OQ is perpendicular to L, L, Q Q is on L. L. P P is on L, L, draw two tangents L1,L2 L_1, L_2 to circle C. C. QA,QB QA, QB are perpendicular to L1,L2 L_1, L_2 respectively. (A A on L1, L_1, B B on L2 L_2). Prove that, line AB AB intersect QO QO at a fixed point. Original formulation: A line l l does not meet a circle ω \omega with center O. O. E E is the point on l l such that OE OE is perpendicular to l. l. M M is any point on l l other than E. E. The tangents from M M to ω \omega touch it at A A and B. B. C C is the point on MA MA such that EC EC is perpendicular to MA. MA. D D is the point on MB MB such that ED ED is perpendicular to MB. MB. The line CD CD cuts OE OE at F. F. Prove that the location of F F is independent of that of M. M.
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