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What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line! See more

Let \(\triangle ABC\) be an acute angled triangle with side lengths \(AB= 5, BC= 7, CA= 8.\) Let \(D\) be the foot of perpendicular from \(A\) to \(BC,\) and let \(O\) be its circumcenter. The feet of perpendiculars from \(O\) to \(AB\) and \(AC\) intersect \(AD\) at points \(Q\) and \(P\) respectively. Let \(S\) be the circumcenter of \(\triangle OPQ.\) If \(\cos (\angle CAS) = a \sqrt{\dfrac{b}{c}}\) for some squarefree coprime positive integers \(b,c\) and a positive integer \(a,\) find \(a+b+c.\)

**Details and assumptions**

- This problem is adapted from an ARO 10th grade geometry problem.

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

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

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In the figure above, all the triangles have integer areas and sides, no two of which are alike. The diameters of the incircles are as shown.

Find the integer diameter of the red incircle of the largest triangle.

**Note**: The answer is not 562.

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Inside \(\triangle ABC\) lies a point \(P\) satisfying \(\angle CAP=15^{\circ}\). Let \(Q\) be another point inside the triangle such that \(\angle QCB=\angle ACP,\angle QBC=\angle ABP\). Suppose the circumcenter of \(ABC\) lies on \(QP\) and \(\angle A=55^{\circ}\), find \(\angle PCB+\angle QBC\) in degrees.

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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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