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Print51st Ukrainian National Mathematical Olympiad, 3rd Round
Ukraine geometry
Problem
Triangle ABC is inscribed into the circle. Tangents at points A and B meet at point T. The line, that passes through the point T and is parallel to AC, intersects BC at point D. Prove that .
Solution
Since triangle is isosceles, and share common arc, then:
This implies, that is cyclic (fig. 9). Hence and triangles and are similar. Triangle is isosceles, thus so is . This implies that .
This implies, that is cyclic (fig. 9). Hence and triangles and are similar. Triangle is isosceles, thus so is . This implies that .
Techniques
TangentsCyclic quadrilateralsAngle chasing