Sign up to access problem solutions.

Already have an account? Log in here.

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

Sign up to access problem solutions.

Already have an account? Log in here.

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 \(BC\).

Sign up to access problem solutions.

Already have an account? Log in here.

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

Sign up to access problem solutions.

Already have an account? Log in here.

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 \(ABCD\).

Sign up to access problem solutions.

Already have an account? Log in here.

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.

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...