MathDB

2

Part of 2019 India Regional Mathematical Olympiad

Problems(2)

Equilateral pair

Source: RMO 2019 P2

10/20/2019
Let ABCABC be a triangle with circumcircle Ω\Omega and let GG be the centroid of triangle ABCABC. Extend AG,BGAG, BG and CGCG to meet the circle Ω\Omega again in A1,B1A_1, B_1 and C1C_1. Suppose ∠BAC=∠A1B1C1,∠ABC=∠A1C1B1\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1 and ∠ACB=B1A1C1 \angle ACB = B_1 A_1 C_1. Prove that ABCABC and A1B1C1A_1 B_1 C_1 are equilateral triangles.
geometryRMOP2
Construct circle ...

Source: P-2 RMO Maharashtra and Goa 2019

11/10/2019
Given a circle τ\tau, let PP be a point in its interior, and let ll be a line through PP. Construct with proof using ruler and compass, all circles which pass through PP, are tangent to τ\tau and whose center lies on line ll.
geometryconstructionconstructive geometry