Construct the triangle ABC with the vertex A being the origin. Then, construct the angle bisector BCA. Next, construct perpendicular lines through B perpendicular to b and through A perpendicular to A. Construct [math]T_b[/math]as the point that intersects the line through D perpendicular to b. Construct the point H (orthocenter). [br]a- as you can see the x coordinate of B is h.[br]b- As you can see the point C =([math]\frac{\left(h^2+kq\right)}{h},0[/math])[br]c- angel bisector theorem shown

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).[br](a) Show that the x-coordinate of B is h.[br](b) Suppose B = (h, q). Show that C is the point [math]\left(\frac{\left(h^2+kq\right)}{h},0\right)[/math][br](c) Use the angle bisector theorem to show that [math]kq^3+h\left(h-2a\right)q^2+\left(h^2-a^2kq\right)+h^2\left(h-a\right)^2=0[/math]