Find co-ordinate of middle point of BC.

Question

Orthocenter and circumcenter of a DABC are $(a, b), (c, d)$. If the co-ordinate of the vertex $A$ are $(x_1, y_1)$ then find co-ordinate of middle point of BC.

Ok the only method I can think of is to take the unknown vertices as $(x_2,y_2)$ and $(x_3,y_3)$ and then find the orthocenter and circumcentre in terms of these vertices.However the typical method is too long.Any shorter way out?

Answer 1

Hint:

Let $a,b,c$ denote as the coordinates of point $A,B,C$ respectively.

Let $h,o$ be the coordinates of orthocenter and circumcenter.

By the Euler's Line theorem, $h+2o = a+b+c$.

The final answer is $\frac{b+c}{2}$