8_3
8_3
Consider the problem of construction of triangle ABC from the vertex A, the orthocenter H, and the trace T_b of the bisector of angle B. Put the origin at A, and let Tb = (a, 0), H = (h, k).
It is known (from Problems I3 and I4) that if B = (h, q), then
(1) Let (a, h, k) = (1, 2, 2).
(a) Show that the equation has no rational roots.
(b) Show that the roots are the intersections of the the rectangular hyperbola xy = 1 and the parabola .
(2) Let (a, h, k) = (1, 2, 1). Show that there is one real solution, which is constructible. Compute the coordinates of the vertices.
(3) Let (a, h, k) = (3, 2, 1). Show that there are three real solutions. One of these has vertices with rational coordinates. Compute the coordinates of the vertices.