

The locus of the intersection of tangents of given angles can be seen from the equation to be a hyperbola.



Take two tangents and their respective normals. Set the parameters of the tangency points to be t and 1/t, and the normals are thus perpendicular.



We see that the subnormal is constant (half the parameter).



The orthocenter of the triangle formed by three tangents to a parabola lies on the directrix. We create the parabola with vertex at the origin and foc...



We see that the ycoordinate is constant, and is in fact the ycoordinate of the directrix. Hence the orthocenter lies on the directrix.



We see that the line joining the intersection to the focus bisects the angle which the tangents’ points of contact make with the focus.



We see that the focus is the midpoint of the segment joining the intersections of the tangent and the normal with the axis:



Given a parabola with vertex A and focus B and a chord CD let E be the intersection of the tangents at C and D. We create the line perpendicular to t...



