Triangle Centers

Triangle Centers: Level 3 Challenges


Two circles \(\omega_1\) and \(\omega_2\) intersect at points \(A\) and \(B\). The tangent to \(\omega_1\) passing through \(A\) intersects \(\omega_2\) at \(X\). The tangent to \(\omega_2\) passing through \(A\) intersects \(\omega_1\) at \(Y\). Let \(O\) be the circumcenter of \(\triangle XAY\). Then what is the measure of \(\angle OBA\) in degrees?

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

If the medians of triangle \(ABC\) intersect at \(O\), then evaluate \( \frac{AB^2+BC^2+AC^2}{OA^2+OB^2+OC^2} \).

Triangle \(ABC\) has centroid \(G\). Triangles \(ABG\), \(BCG\), and \(CAG\) have centroids \(G_1,G_2,G_3\) respectively.

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

Find \(p+q\).

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

\([ABCD]\) denotes the area of \(ABCD\).

How many triangles \(ABC\) are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of \(\triangle ABC\) is \(44\) square units?

Details and assumptions:

The orthocenter of \(ABC\) is the point at which the altitudes of \(ABC\) intersect.

The circumcenter of \(ABC\) is the point which is equidistant from \(A\), \(B\) and \(C\).

The centroid of \(ABC\) is the point at which the medians of \(ABC\) intersect.


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