Question

Let O and H be the circumcenter and orthocenter of triangle ABC, respectively. Let a, b, and c denote the side lengths, and let R denote the circumradius. Find OH^2 if R = 7 and a^2 + b^2 + c^2 = 29.

Answer 1

OH²=412

Explanation:

A Circumcenter of a triangle is the point where the three perpendicular bisectors of a triangle meet.

The Orthocenter is the point where all three altitudes of the triangle intersect.

Given any triangle, the orthocenter, circumcenter and centroid are collinear on the Euler line. By Euler's Identity, the distance between the circumcenter and the orthocenter of a triangle is related to the circumradius(R) and the sides(a,b,c) of the triangle by the relation

OH²=9R²-(a²+b²+c²)

R=7, a²+b²+c²=29

OH²=9(7)²-29

OH²=412