MathDB

6

Part of 2015 Costa Rica - Final Round

Problems(2)

NK // ML wanted, trapezoid and 2 circles with diameters legs related

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 3.6 day2

9/30/2021
Given the trapezoid ABCDABCD with the BC∥ADBC\parallel AD, let C1C_1 and C2C_2 be circles with diameters ABAB and CDCD respectively. Let MM and NN be the intersection points of C1C_1 with ACAC and BDBD respectively. Let KK and LL be the intersection points of C2C_2 with ACAC and BDBD respectively. Given M≠AM\ne A, N≠BN\ne B, K≠CK\ne C, L≠DL\ne D. Prove that NK∥MLNK \parallel ML.
geometrytrapezoidparallel
reflection of BC wrt PQ is tangent to (APQ), <POB = <ABC , <QOC = <ACB

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 3.6

9/23/2021
Let △ABC\vartriangle ABC be a triangle with circumcenter OO. Let P P and QQ be internal points on the sides ABAB and ACAC respectively such that ∠POB=∠ABC\angle POB = \angle ABC and ∠QOC=∠ACB\angle QOC = \angle ACB. Show that the reflection of line BCBC over line PQPQ is tangent to the circumcircle of triangle △APQ\vartriangle APQ.
geometryreflectiontangentequal angles