MathDB

7

Part of 2009 239 Open Mathematical Olympiad

Problems(2)

The Feuerbach point of ABC

Source: 239 2009 S7

7/29/2020
The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle ABCABC) FF is marked in triangle ABCABC. A1A_1 is on the side BCBC such that AA1AA_1 is the altitude of triangle ABCABC. Prove that the line symmetric to FA1FA_1 with respect to BCBC is perpendicular to IOIO, where OO is the center of the circumcircle of the triangle ABCABC and II is the center of its incircle.
geometry
Angle inequality

Source: 239 2009 J7

7/29/2020
In the triangle ABCABC, the cevians AA1AA_1, BB1BB_1 and CC1CC_1 intersect at the point OO. It turned out that AA1AA_1 is the bisector, and the point OO is closer to the straight line ABAB than to the straight lines A1C1A_1C_1 and B1A1B_1A_1. Prove that ∠BAC>120∘\angle{BAC} > 120^{\circ}.
inequalities