MathDB

6

Part of 2020 Tuymaada Olympiad

Problems(2)

Angle chasings

Source: 2020 Tuymaada Junior P6

10/6/2020
AKAK and BLBL are altitudes of an acute triangle ABCABC. Point PP is chosen on the segment AKAK so that LK=LPLK=LP. The parallel to BCBC through PP meets the parallel to PLPL through BB at point QQ. Prove that ∠AQB=∠ACB\angle AQB = \angle ACB.
(S. Berlov)
geometry
Circumcenter on an isosceles triangle.

Source: Tuymaada 2020 Senior, P6

10/6/2020
An isosceles triangle ABCABC (AB=BCAB = BC) is given. Circles ω1\omega_1 and ω2\omega_2 with centres O1O_1 and O2O_2 lie in the angle ABCABC and touch the sides ABAB and CBCB at AA and CC respectively, and touch each other externally at point XX. The side ACAC meets the circles again at points YY and ZZ. OO is the circumcenter of the triangle XYZXYZ. Lines O2OO_2 O and O1OO_1 O intersect lines ABAB and BCBC at points C1C_1 and A1A_1 respectively. Prove that BB is the circumcentre of the triangle A1OC1A_1 OC_1.
geometrycircumcircleTriangle