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Triangle Centers
What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line! See more
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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by
Sharky Kesa
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.
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by
Bolin Chen
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.
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by
samardeep sarna
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: make no assumptions based on the diagram
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by
Daniel Liu
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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by
Sharky Kesa