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Print74th NMO Selection Tests for JBMO
Romania geometry
Problem
Let be a triangle. An arbitrary circle which passes through the points and intersects the sides and for the second time in and , respectively. The line intersects the circumcircle of triangle at and , and the line intersects the circumcircle of the triangle at and , such that is situated on the segment , and lies on the segment . Prove that:
a) the points and are concyclic;
b) the triangle is isosceles.
a) the points and are concyclic;
b) the triangle is isosceles.
Solution
a) Denote by the intersection of the lines and . Writing the power of with respect to all three circles, we obtain: Since , we infer that the points and are concyclic.
b) The triangles and are similar (A.A.), thus . Similarly, we find that .
From the power of the point with respect to the circumcircle of the quadrilateral , we infer that , therefore .
Similarly, we prove that , therefore the triangle is isosceles.
b) The triangles and are similar (A.A.), thus . Similarly, we find that .
From the power of the point with respect to the circumcircle of the quadrilateral , we infer that , therefore .
Similarly, we prove that , therefore the triangle is isosceles.
Techniques
Radical axis theoremCyclic quadrilateralsAngle chasingDistance chasing