TITLE: Given A and B two fixed points on the circle find the locus of the orthocenter of triangle ABC where C is a mobile point on the circle.
QUESTION [1 upvotes]: First we have to find the locus and then we have to prove double inclusion.
Please help me !

REPLY [1 votes]: Due to Euler's theorem, the circumcenter, orthocenter and centroid of a triangle are always collinear. Moreover, $HG=2\cdot GO$ holds. This means that if we take the circumcenter $O$ as the origin of $\mathbb{R}^2$, the vector identities
$$ G = \frac{A+B+C}{3},\qquad H=A+B+C $$
hold. This means that if $A,B$ stay fixed while $C$ runs over the circumcircle $\Gamma$, $H$ describes a circle congruent to $\Gamma$ having $A+B$ as center. The converse implication is straightforward since $\{A,B,C,H\}$ is an orthocentric system, giving that $C$ is the orthocenter of $ABH$.
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