BF is perpendicular to diameter CD. E is the intersection point of tangents at B and D. Observe that BG = FG.
Can you prove it?
This is from the Liber Assumptorum (Book of Lemmas) translated from Arabic but attributed to Archimedes.

A theorem of Archimedes on a circle can be generalized to ellipses as shown here.
CE is a diameter of the ellipse. FG is parallel to the tangent at C. H is the intersection of the tangents at F and C.
Observe that EH bisects FG.

A theorem of Archimedes on a circle can be generalized to a hyperbola as shown here.
CE is a diameter of the hyperbola. FG is parallel to the tangent at C. H is the intersection of the tangents at F and C.
Observe that EH bisects FG.

A theorem of Archimedes on a circle can be generalized to parabolas as shown here.
Vertical lines correspond to diameters of the parabola. FG is parallel to the tangent at C. E is the intersection of the tangents at F and C.
Observe that teh vertical line through E bisects FG.