Question

A circle cuts a rectangualar hyperbola xy = c² at A, B, C and D. If H is the orthocentre of ΔABC, then show that H and D are the extremities of a diameter of the curve.

Answer 1

If (ct_r ,c/t_r) (where r = 1, 2, 3 and 4) are points A, B, C and D, respectively, the orthocentre H of triangle ABC is

which also lies on the curve. Now, since t₁t₂t₃t₄ , we have

t₄ = 1/t₁t₂t₃

so that

and hence (0, 0) is the midpoint of HD.