MathDB

3

Part of 1999 All-Russian Olympiad

Problems(3)

Incenter lies on common tangent

Source: All-Russian MO 1999

12/31/2012
A triangle ABCABC is inscribed in a circle SS. Let A0A_0 and C0C_0 be the midpoints of the arcs BCBC and ABAB on SS, not containing the opposite vertex, respectively. The circle S1S_1 centered at A0A_0 is tangent to BCBC, and the circle S2S_2 centered at C0C_0 is tangent to ABAB. Prove that the incenter II of △ABC\triangle ABC lies on a common tangent to S1S_1 and S2S_2.
geometryincentergeometry unsolved
rhombus

Source: All-Russian MO 1999

12/22/2011
A circle touches sides DADA, ABAB, BCBC, CDCD of a quadrilateral ABCDABCD at points KK, LL, MM, NN, respectively. Let S1S_1, S2S_2, S3S_3, S4S_4 respectively be the incircles of triangles AKLAKL, BLMBLM, CMNCMN, DNKDNK. The external common tangents distinct from the sides of ABCDABCD are drawn to S1S_1 and S2S_2, S2S_2 and S3S_3, S3S_3 and S4S_4, S4S_4 and S1S_1. Prove that these four tangents determine a rhombus.
geometryrhombusgeometric transformationreflectionparallelogramgeometry unsolved
Concurrent Tangent

Source: All-Russian MO 1999

1/10/2011
The incircle of △ABC\triangle ABC touch ABAB,BCBC,CACA at KK,LL,MM. The common external tangents to the incircles of △AMK\triangle AMK,△BKL\triangle BKL,△CLM\triangle CLM, distinct from the sides of △ABC\triangle ABC, are drawn. Show that these three lines are concurrent.
geometryincentergeometric transformationreflectiongeometry proposed