A triangle ABC is inscribed in the circle x^2 + y^2 - 4x - 6y - 23 = 0 such that BAC = 2 . If H is the orthocenter and G is the centroid of the ABC , then the length of the line segment HG is equal to:

Step-by-step solution

The given circle is x^2 + y^2 - 4x - 6y - 23 = 0 . The center of the circle is O(2, 3) and its radius is r = (-2)^2 + (-3)^2 - (-23) = 4 + 9 + 23 = 36 = 6 . Since BAC = 2 , the triangle ABC is a right-angled triangle with the right angle at vertex A . In a right-angled triangle, the orthocenter H coincides with the vertex containing the right angle. Thus, H is at A . The circumcenter of the right-angled triangle is the midpoint of its hypotenuse BC . Since the triangle is inscribed in the given circle, the circumce