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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 4 Challenges

Three circles of equal radii all intersect at a single point $$P$$. Let the other intersections be $$A$$, $$B$$ and $$C$$. Which of the following must be true?

In triangle $$ABC$$, $$O$$ is the circumcenter and $$I$$ is the incenter. $$P$$ is a point on the exterior angle bisector of $$\Delta BAC$$ such that $$PI \perp OI$$. Extend $$PI$$ to intersect $$BC$$ at $$Q$$.

Find $$\dfrac{PI}{QI}$$.

Note: Figure not drawn to scale.

If the vertices of a triangle have rational coordinates, then the coordinates of which of the following are necessarily rational?

(A) Centroid
(B) Circumcenter
(C) Orthocenter
(D) Incenter

Triangle $$AOB$$ and triangle $$BOC$$ are directly similar (they have the same orientation). Let the centroids of these two triangles be $$M_1$$ and $$M_2$$, respectively.

If triangle $$M_1OM_2$$ is right, and $$M_1M_2=AB$$, then the sum of all possible values of $$\cos^2(\angle ABO)$$ can be expressed as $$\dfrac{p}{q}$$ for positive coprime integers $$p,q$$.

Find $$p+q$$.

##### Note: We do not know which angle of $$\triangle M_1 O M_2$$ is the right angle.

Three circles of equal radii all intersect at a single point $$P$$. Let the other intersections be $$A$$, $$B$$ and $$C$$. Which of the following must be true?

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