Geometry
# Triangle Centers

$\Delta ABC$ is inscribed in a circle of radius $5$. $H$ is the orthocentre of $\Delta ABC$. If $\tan A = 1$, $\tan B = 2$ then find $CH^{2}$.

Triangle $ABC$ has centroid $G$. Triangles $ABG$, $BCG$, and $CAG$ have centroids $G_1,G_2,G_3$ respectively.

The value of $\dfrac{[G_1G_2G_3]}{[ABC]}$ can be represented by $\dfrac{p}{q}$, for positive coprime integers $p,q$.

Find $p+q$.

$\text{Details and Assumptions:}$

$[ABCD]$ denotes the area of $ABCD$.

How many triangles $ABC$ are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of $\triangle ABC$ is $44$ square units?

**Details and assumptions:**

The orthocenter of $ABC$ is the point at which the altitudes of $ABC$ intersect.

The circumcenter of $ABC$ is the point which is equidistant from $A$, $B$ and $C$.

The centroid of $ABC$ is the point at which the medians of $ABC$ intersect.