Triangle Centers: Level 4 Challenges Practice Problems Online

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

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

Intro

Concept Quizzes

Challenge Quizzes

Triangle Centers: Level 4 Challenges

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

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Triangle Centers

Intro

Concept Quizzes

Challenge Quizzes

Triangle Centers: Level 4 Challenges

Note: We do not know which angle of △M1OM2 \triangle M_1 O M_2 △M1​OM2​ is the right angle.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

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