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PrintChina Mathematical Olympiad
China geometry
Problem
Suppose points and are the incenter and orthocenter of an acute triangle respectively, and points and are the midpoints of sides and respectively. It is known that ray intersects side at (), and ray intersects the extension of at : and intersect at and is the circumcenter of . Prove that three points , and are collinear if and only if the areas of and are equal. (posed by Shen Wenxuan)
Solution
Proof First, we will prove that three points , and are collinear . As shown in the figure, assume that is the circumcenter of . We join and , then
Secondly, we will prove .
Construct at point , and at , then Note that , therefore . Assume , where is the radius of the inscribed circle of . Then . Furthermore set , and , then . From and , we have , so . Similarly, . Hence,
Secondly, we will prove .
Construct at point , and at , then Note that , therefore . Assume , where is the radius of the inscribed circle of . Then . Furthermore set , and , then . From and , we have , so . Similarly, . Hence,
Techniques
Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleTriangle trigonometryAngle chasingConstructions and loci