MathDB

10

Part of 2007 QEDMO 4th

Problems(1)

Not only X on A-altitude, but also AO_aA_aX parallelogram

Source: 4th QEDMO problem 9, originally from John Sturgeon Mackay in Edinburgh Proceedings 1883

11/10/2007
Let ABC ABC be a triangle. The A A-excircle of triangle ABC ABC has center Oa O_{a} and touches the side BC BC at the point Aa A_{a}. The B B-excircle of triangle ABC ABC touches its sidelines AB AB and BC BC at the points Cb C_{b} and Ab A_{b}. The C C-excircle of triangle ABC ABC touches its sidelines BC BC and CA CA at the points Ac A_{c} and Bc B_{c}. The lines CbAb C_{b}A_{b} and AcBc A_{c}B_{c} intersect each other at some point X X. Prove that the quadrilateral AOaAaX AO_{a}A_{a}X is a parallelogram. Remark. The A A-excircle of a triangle ABC ABC is defined as the circle which touches the segment BC BC and the extensions of the segments CA CA and AB AB beyound the points C C and B B, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle CAB CAB and the exterior angle bisectors of the angles ABC ABC and BCA BCA. Similarly, the B B-excircle and the C C-excircle of triangle ABC ABC are defined. [hide="Source of the problem"]Source of the problem: Theorem (88) in: John Sturgeon Mackay, The Triangle and its Six Scribed Circles, Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.
geometryparallelogramratioangle bisectorexterior anglegeometry proposed