Triangle Centers

Triangle Centers: Level 4 Challenges


Three circles of equal radii all intersect at a single point PP. Let the other intersections be AA, BB and CC. 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 AOBAOB and triangle BOCBOC are directly similar (they have the same orientation). Let the centroids of these two triangles be M1M_1 and M2M_2, respectively.

If triangle M1OM2M_1OM_2 is right, and M1M2=ABM_1M_2=AB, then the sum of all possible values of cos2(ABO)\cos^2(\angle ABO) can be expressed as pq\dfrac{p}{q} for positive coprime integers p,qp,q.

Find p+qp+q.

Note: We do not know which angle of M1OM2 \triangle M_1 O M_2 is the right angle.

Let ABCABC be a triangle in which AB=ACAB=AC. Suppose the orthocentre of the triangle lies on the incircle.

Find ABBC\dfrac{AB}{BC}.

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

Let ADBC,BEAC,CFABAD \perp BC, BE \perp AC, CF \perp AB and let HH be its orthocenter

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

Let r0r_0 be inradius and r1,r2,r3r_1, r_2, r_3 be the exradii of ΔABC \Delta ABC

Then find the value of

R(ABC)+R(HAB)+R(HBC)+R(HAC)+i=03ri R(ABC) + R(HAB) + R(HBC) + R(HAC) + \sum_{i=0}^3 r_i


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