Let C be a point on a hyperbola. Let CB be the perpendicular from that point to the diameter. Let G and H be the intersections of the diameter with the curve, and choose A on the diameter, or the diameter extended, so that AH: AG = BH: BG. Then AC will be tangent to the curve at C.

Hint: In modern notation, write the equation x ² − y ² = 1 for the hyperbola. Denote a ² b ²

H=(a,0),G=(−a,0),a>0,B=(−c,0),c>0. ThenfindC,findtheequationof the tangent line of the hyperbola at C, and find the point A