Find the equation of the third side of the triangle given two equations of sides and orthocenter.

Question

Orthocenter lies on the origin of the triangle and the equation of 2 sides of the triangle are x+y-4=0 and 2x+y-1=0. What is the equation of third side?

Answer 1

I did not find an easy way to solve this problem.   I did find a solution via the standard diagramatic proof that the three altitudes of a triangle are concurrent coupled with a brute force application of coordinate geometry.   The method requires pages of algebra - too much for this space.  

 

The answer is that the equation of the third side is y =(3/7) x + 4/21

 

The three triangle vertices are   ( -3, 7) ; (8/3 , 4/3)  and (1/3 , 1/3)