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Archive for Tháng Hai, 2010

Let (I) and (O) be the incircle and circumcircle of triangle ABC, respectively, P be an arbitrary point on (O). Construct two tangents from P to (I), they cut BC at E and F. Prove that (PEF) passes through the tangency of A-Mixtilinear incircle of triangle ABC and (O).

Proof: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=186117&hilit=Mixtilinear

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Given triangle ABC and its circumcenter O. Let P be an arbitrary point on the plane, A_1B_1C_1 be the pedal triangle of P wrt \Delta ABC. (A_1B_1C_1) intersects 9-point circle of triangle ABC at X_1 and X_2. By Fontene’s theorem, we know that X_1 is the Anti-Steiner point of OP wrt median triangle of triangle ABC. Let P' be the isogonal conjugate of P wrt \Delta ABC, M be the midpoint of AP', M_a be the midpoint of BC, H be the projection of A onto BC. (MX_1H)\cap BC=\{H,E\}, (MX_1H)\cap (A_1B_1C_1)=\{F\}. Prove that the intersection of EF and M_aX_2 lies on (A_1B_1C_1).

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A property of Anti-Steiner point 2

Given triangle ABC and its circumcircle (O). Let P be an arbitrary point in the plane. AP, BP, CP cuts (O) again at A_1, B_1, C_1, respectively. Let A_2, B_2, C_2 be the reflections of P wrt \Delta ABC. Prove that 4 circles (O), (PA_1A_2), (PB_1B_2), (PC_1C_2) are concurrent.

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Given triangle  ABC with an arbitrary point P. Prove that the Cevian circle, the Pedal circle of P and the 9-point circle of triangle ABC are concurrent.

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