MathDB

2

Part of 2018 China Second Round Olympiad

Problems(2)

Equal lengths in incircle configuration

Source: China second round 2018 (A) Q2

5/5/2019
In triangle △ABC\triangle ABC, AB<ACAB<AC, M,D,EM,D,E are the midpoints of BCBC, the arcs BACBAC and BCBC of the circumcircle of △ABC\triangle ABC respectively. The incircle of △ABC\triangle ABC touches ABAB at FF, AEAE meets BCBC at GG, and the perpendicular to ABAB at BB meets segment EFEF at NN. If BN=EMBN=EM, prove that DFDF is perpendicular to FGFG.
geometryincentercircumcirclecongruent triangles
\frac{AD}{DC}=\frac{BC}{2CE}

Source: China second round 2018 (B) Q2

6/22/2019
In triangle △ABC,AB=AC.\triangle ABC, AB=AC. Let DD be on segment ACAC and EE be a point on the extended line BCBC such that CC is located between BB and EE and ADDC=BC2CE\frac{AD}{DC}=\frac{BC}{2CE}. Let ω\omega be the circle with diameter AB,AB, and ω\omega intersects segment DEDE at F.F. Prove that B,C,F,DB,C,F,D are concyclic.
geometry