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Romania geometry
Problem
In a triangle denote by , , respectively , the points where the angle bisectors of , , respectively , meet its circumcircle.
a) Prove that the ortocenter of triangle coincides with the incenter of triangle .
b) Prove that if , then the triangle is equilateral.
a) Prove that the ortocenter of triangle coincides with the incenter of triangle .
b) Prove that if , then the triangle is equilateral.
Solution
a) Let be the incenter of the triangle . are the midpoints of the arcs . The angle determined by the lines and is equal to , hence . Likewise, , and the claim follows.
b) Let be the circumcenter of the triangle . The given relation implies that . Invoking Sylvester's theorem, the triangles and share the same orthocenter. Thus, in the triangle point is both incenter and orthocenter, yielding the conclusion.
b) Let be the circumcenter of the triangle . The given relation implies that . Invoking Sylvester's theorem, the triangles and share the same orthocenter. Thus, in the triangle point is both incenter and orthocenter, yielding the conclusion.
Techniques
Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleAngle chasingVectors