MathDB

G1

Part of 2020-IMOC

Problems(1)

IMOC 2020 G1 (tangent circles wanted)

Source: https://artofproblemsolving.com/community/c6h2254883p17398793

9/1/2020
Let OO be the circumcenter of triangle ABCABC. Choose a point XX on the circumcircle ⊙(ABC)\odot (ABC) such that OX∥BCOX\parallel BC. Assume that ⊙(AXO)\odot(AXO) intersects AB,ACAB, AC at E,FE, F, respectively, and OE,OFOE, OF intersect BCBC at P,QP, Q, respectively. Furthermore, assume that ⊙(XPQ)\odot(XP Q) and ⊙(ABC)\odot (ABC) intersect at RR. Prove that OROR and ⊙(XPQ)\odot (XP Q) are tangent to each other.
(ltf0501)
circumcircletangent circlesgeometry