MathDB

4

Part of 2024 Tuymaada Olympiad

Problems(2)

T,H,E,B are concyclic

Source: Tuymaada 2024 Junior P4

7/10/2024
A triangle ABCABC is given. NN and MM are the midpoints of ABAB and BCBC, respectively. The bisector of angle BB meets the segment MNMN at EE. HH is the base of the altitude drawn from BB in the triangle ABCABC. The point TT on the circumcircle of ABCABC is such that the circumcircles of TMNTMN and ABCABC are tangent. Prove that points T,H,E,BT, H, E, B are concyclic. Proposed by M. Yumatov
geometry
Interesting Excircles Problem

Source: Tuymadaa Senior P4 2024

7/7/2024
A triangle ABCABC is given. The segment connecting the points where the excircles touch ABAB and ACAC meets the bisector of angle CC at XX. The segment connecting the points where the excircles touch BCBC and ACAC meets the bisector of angle AA at YY. Prove that the midpoint of XYXY is equidistant from AA and CC.
tuymadaageometryexcirclesmidpoint