Summary: 62 2. Elementary geometrical figures and their properties
Proof. Clearly, the second and third inequalities can be derived from
the first by a renumbering of points, so we show only the first one.
Both sides of the first inequality are translation invariant. Because of
this we can, without loss of generality, assume z1 = 0. The triangle
|z2z3 - z2z4| |z2z3 - z3z4| + |z3z4 - z2z4|
and can be written equivalently as
|z2||z3 - z4| |z3||z2 - z4| + |z4||z2 - z3|.
This is precisely the inequality d12d34 d13d24 + d23d14. If we have
equality then there exists a real number with
(z2z3 - z3z4) = (z3z4 - z2z4).
Since z1 = 0 the number - is the cross ratio of the four points. In
other words, equality (in any of the three inequalities) occurs if and
only if the cross ratio of the four points is real. The assertion about
the case of equality is a consequence of Corollary 9.
2.4. The conic sections
2.4.1. The section by a plane of a cone. Consider a cone C with
vertex S = (0, 0, 0), the origin, and half opening angle /2 - . The
equation for C in R3