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prove that two circles and a line have a common point

Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 4

October 12, 2024
geometrytangent

Problem Statement

Let ABC ABC be an acute-angled scalene triangle. Let D D be a point on the interior of segment BC BC , different from the foot of the altitude from A A . The tangents from A A and B B to the circumcircle of triangle ABD ABD meet at O1 O_1 , and the tangents from A A and C C to the circumcircle of triangle ACD ACD meet at O2 O_2 . Show that the circle centered at O1 O_1 passing through A A , the circle centered at O2 O_2 passing through A A , and the line BC BC have a common point.
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