Two circles and intersect at points and . The tangent to passing through intersects at . The tangent to passing through intersects at . Let be the circumcenter of . Then what is the measure of in degrees?
Let be an acute triangle with . Let be the orthocenter (intersection of altitudes) of this triangle and be the midpoint of . The ray intersects the circumcircle of triangle at point , where belongs to the minor arc . It is known that , and . Find the length of .
If the medians of triangle intersect at , then evaluate .
Triangle has centroid . Triangles , , and have centroids respectively.
The value of can be represented by , for positive coprime integers .
Find .
denotes the area of .
How many triangles are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of is square units?
Details and assumptions:
The orthocenter of is the point at which the altitudes of intersect.
The circumcenter of is the point which is equidistant from , and .
The centroid of is the point at which the medians of intersect.