What will be the other vertex of the triangle? Two vertices of a triangles are $(5,-1)$ and $(-2,3)$. If the orthocenter of the triangle is the origin, what is the other vertex ? My approach was that since the three vertices and the orthocenter form an orthic system, I need to find the orthocenter of the triangle with the three given points. Thus make two equations and solve... but it is becoming too cumbersome,is there any better method ?

(I assume that you meant "the orthocenter of the _triangle_ is the origin.")

Let $A$ be $(5,-1)$, $B$ be $(-2,3)$, $O$ be the origin, and your desired third vertex be $C$.

Find the line through $O$ perpendicular to $\overline {AB}$: this will be the altitude of the triangle through $C$. (I get $-7x+4y=0$.) Then find the line through $A$ perpendicular to $\overline {BO}$: this will define the side of the triangle opposite to $B$, so this line also goes through $C$. (I get $-2x+3y=13$.)

The intersection of those two lines is $C$, your third vertex of the triangle. (I find the point to be $(-4,-7)$.)

You can check your answer by seeing that the line through $A$ and $O$ is perpendicular to the line through $B$ and $C$.

I'm sure you know the easy way to find perpendicular lines and to check that two lines are perpendicular.