I have the following scenario.
The coordinates of points B and D are $(10,0)$ and $(10,-10)$ respectively.
I want to construct angle $\angle BFD = 45^\circ $. How can i find the coordinates of point F in order to construct this angle? In other words how can i find the equations of the lines BF and FD. Is it even possible with the data given? Thanks in advance.
First, construct the perpendicular bisector of $BD$, $\ell$, let $M$ the point where $\ell$ intersects $BD$, let $P$ the point where the circle centered at $M$ by $B$ and $D$ intersects $\ell$ (see the picture below). Then, trace a circle centered at $P$ passing by $B$ and $D$. You can show that any point $F$ on the larger arc $BC$ works.