Geometry

Triangle Centers

Triangle Centers: Level 3 Challenges

         

Two circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB. The tangent to ω1\omega_1 passing through AA intersects ω2\omega_2 at XX. The tangent to ω2\omega_2 passing through AA intersects ω1\omega_1 at YY. Let OO be the circumcenter of XAY\triangle XAY. Then what is the measure of OBA\angle OBA in degrees?

Let ABCABC be an acute triangle with AB>BCAB > BC. Let HH be the orthocenter (intersection of altitudes) of this triangle and MM be the midpoint of ACAC. The ray MH \overrightarrow{M H} intersects the circumcircle of triangle ABCABC at point PP, where PP belongs to the minor arc BCBC. It is known that ABP=90 \angle ABP=90^{\circ}, MH=5MH=5 and HP=16HP=16. Find the length of BCBC.

If the medians of triangle ABCABC intersect at OO, then evaluate AB2+BC2+AC2OA2+OB2+OC2 \frac{AB^2+BC^2+AC^2}{OA^2+OB^2+OC^2} .

Triangle ABCABC has centroid GG. Triangles ABGABG, BCGBCG, and CAGCAG have centroids G1,G2,G3G_1,G_2,G_3 respectively.

The value of [G1G2G3][ABC]\dfrac{[G_1G_2G_3]}{[ABC]} can be represented by pq\dfrac{p}{q}, for positive coprime integers p,qp,q.

Find p+qp+q.

Details and Assumptions:\text{Details and Assumptions:}

[ABCD][ABCD] denotes the area of ABCDABCD.

How many triangles ABCABC are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of ABC\triangle ABC is 4444 square units?


Details and assumptions:

The orthocenter of ABCABC is the point at which the altitudes of ABCABC intersect.

The circumcenter of ABCABC is the point which is equidistant from AA, BB and CC.

The centroid of ABCABC is the point at which the medians of ABCABC intersect.

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