This proof appears on DIOCLES, On Burning Mirrors by Toomer, G. J p.10
The proof starts in the paragraph starting with "To explain what i mean..." and ends in that same page. The only figure to pay atention to is figure 3 (the only one in the image), figures 4 and 5 are not relevant (they show the same proof for the hyperbola and ellipse)
The thing is, I follow the proof perfectly up to and after where it says $HZ=2ZF$, I don't understand where does that statement come from. Since the author does not bother to explain it I guess it's trivial but I have been staring at my screen for an hour and I just dont see it
Can anyone explain to me why $HZ=2ZF$ ?
Call W the point of intersection of GZ with the axis of the cone AF. Since GZ is a dimeter of a circle parallel to the circle MKN, W is the midpoint of GZ. Next, observe that the triangles ZWF and ZGH are similar (both right triangles with a common acute angle) Therefore, ZF=FH (because ZW=WG).