Given an ellipse, we want to draw a set of tangents to it, which are orthogonal to each other. We are looking for a synthetic way; not through algebra or trigonometry.
I know how to construct one tangent: Given any point $P$ on the ellipse, we connect it with the two foci, say $F_1$ and $F_2$ and then we bisect angle $F_1PF_2$. The bisector is the "normal". Then we draw an orthogonal line to the bisector, from point $P$f and this is our tangent.
I am trying to think of a way to construct another tangent, orthogonal to it. Clearly the normal of the second tangent will be orthogonal to the 1st normal. We are looking for the single point $Q$ (or maybe 2 points), for which, the 2nd normal will bisect angle $F_1QF_2$ but I can't think of any way to construct it.
Any ideas? Thank you!