In triangle $ABC$, let $A(3,4)$ and the equation of angle bisector of $B$ is $y=x$. If orthocenter of triangle is $(2,2)$ and $B(h,k)$, then find $(h,k)$.
I cant solve this question, I am confused, what is the use of equation of angle bisector? Some hints will be helpful for me.
B lies on its own angle bisector so $k=h$. The orthocentre $H(2,2)$ also lies on this line, so by symmetry the triangle ABC is isosceles with $BA=BC$. By symmetry also, we can locate $ C $ at $(4,3)$ (a picture helps).
Now we require that $AH$ is perpendicular to $BC$,.so using the product of gradients rule, we have $$\frac{2-4}{2-3}\times \frac{h-3}{h-4}=-1$$
Thus $h=k=\frac{10}{3}$
