Geometry

Triangle Centers

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?

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.

Let \(ABC\) be a triangle in which \(AB=AC\). Suppose the orthocentre of the triangle lies on the incircle.

Find \(\dfrac{AB}{BC}\).

Consider \( \Delta ABC \) such that \(AB = 13, BC = 14, AC = 15 \)

Let \(AD \perp BC, BE \perp AC, CF \perp AB \) and let \(H\) be its orthocenter

Let \(R(ABC)\) denote circumradius of \( \Delta ABC \)

Let \(r_0 \) be inradius and \(r_1, r_2, r_3 \) be the exradii of \( \Delta ABC \)

Then find the value of

\[ R(ABC) + R(HAB) + R(HBC) + R(HAC) + \sum_{i=0}^3 r_i \]

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