Co-ordinate Geometry (Triangles)(Orthocenter)

Question

Is there any way to find out the equation of orthocenter of a triangle, when equation of the three sides of a triangle are given, without finding out the vertices of the triangle?

Answer 1

If the sides are $L_1 = 0, L_2 = 0, L_3 = 0$, we can find the altitude through the vertices as follows: Find $k$ such that $L_1 + kL_2 = 0$ is perpendicular to $L_3 = 0$. This will give the altitude through the vertex that is the intersection of $L_1 = 0$ and $L_2 = 0$. Similarly find one more altitude and the point of intersection of these altitudes is the orthocenter.