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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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