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# Triangle Centers

What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line!

# Triangle Centers: Level 3 Challenges

Two circles $$\omega_1$$ and $$\omega_2$$ intersect at points $$A$$ and $$B$$. The tangent to $$\omega_1$$ passing through $$A$$ intersects $$\omega_2$$ at $$X$$. The tangent to $$\omega_2$$ passing through $$A$$ intersects $$\omega_1$$ at $$Y$$. Let $$O$$ be the circumcenter of $$\triangle XAY$$. Then what is the measure of $$\angle OBA$$ in degrees?

Let $$ABC$$ be an acute triangle with $$AB > BC$$. Let $$H$$ be the orthocenter (intersection of altitudes) of this triangle and $$M$$ be the midpoint of $$AC$$. The ray $$\overrightarrow{M H}$$ intersects the circumcircle of triangle $$ABC$$ at point $$P$$, where $$P$$ belongs to the minor arc $$BC$$. It is known that $$\angle ABP=90^{\circ}$$, $$MH=5$$ and $$HP=16$$. Find the length of $$BC$$.

If the medians of triangle $$ABC$$ intersect at $$O$$, then evaluate $$\frac{AB^2+BC^2+AC^2}{OA^2+OB^2+OC^2}$$.

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.

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