prove tolmey's theorem

abhilash

Ptolemy's Relation

can be reformulated as

valid for the cross-ratio

of any four (pairwise different) complex numbers $z_1,\ldots,z_4$ .

To make this connection explicit one represents the four vertices $P_1, \ldots, P_4$ as four complex numbers $z_1, \ldots, z_4$ of norm one, arranged in (counterclockwise) cyclic order on the unit circle. For two complex numbers $x, y$ on the unit circle their squared distance equals

Therefore for any quadruple of (pairwise different) complex numbers $(z 1, z 2, z 3, z 4)$ on the unit circle the square of the "length

cross-ratio"

is equal to the square $\,\mbox{cr}^2(z_1,z_2,z_3,z_4)$ of the ordinary ("complex points" ) cross-ratio

. Taking square roots one first deduces

for any quadruple $\,(z_1,\ldots,z_4)$ of points on the unit circle. The sign factor $\epsilon \in \{ -1,1\}$ depends on the relative position of the four points $\,z_1, \ldots, z_4$ on the unit circle and can be determined using the invariance of the cross-ratio under a linear fractional transformation $z \mapsto {{az+b}\over{cz+d}}$ . Assume that the quadruple $\,(z_1, \ldots,z_4)$ on the unit circle is arranged in natural (counterclockwise) cyclic order. Then

$\,\mbox{cr}(z_1,z_2,z_3,z_4)>1.$

This property can be proved using the projective transformation $r:\; z \mapsto i{{(1+z)}\over{(1-z)}}$ (which is the "inverse Cayley transform "). It maps the punctured unit circle $S^{1}\setminus \{z=1\}$ (continuously) to the real line $\mathbb{R}$ (with the upper (resp. lower) arc of the unit circle mapping to the negative (resp. positive) half-line). In polar coordinates the map is given as $\,r(e^{ i \alpha})=-\cot( \alpha /2)$ which shows that it defines a monotone function in the "angle" coordinate $\alpha \in )0,2\pi($ . Therefore the sign of the cross-ratio can be read off from the mutual order of the image points on the real line. After multiplying the $z i$ with a suitable scalar $z'$ of norm 1 one may in addition assume that $z_i \ne 1$ for all $i$ . If the quadruple $(\,z_1, \ldots,z_4)$ on the unit circle (punctured at $z = 1$ ) is arranged in natural (counterclockwise) cyclic order the image quadruple $\,(y_1, \ldots, y_4):=(\,r(z_1),r(z_2),r(z_3),r(z_4)\,)$ satisfies $\, y_1 <y_2 <y_3<y_4$ . The relation

then shows that $\,\mbox{cr}(z_1,z_2,z_3,z_4)=\mbox{cr}(y_1,y_2,y_3,y_4)>1$ . On the other hand, if one interchanges the middle pair $(z 2, z 3)$ in a cyclically ordered quadruple then the cross-ratio will become negative because $\,\mbox{cr}(z_1,z_3,z_2,z_4)=1-\mbox{cr}(z_1,z_2,z_3,z_4)<0$ , using the relation of cross-ratio's

Summarizing the sign discussion one obtains that for a quadruple $(z_1, \ldots, z_4)$ of (pairwise different) points on the unit circle given in (counterclockwise) cyclic order one has

and

Ptolemy's relation

can now be interpreted as the algebraic relation (already used above) between cross-ratios

using the representation of the vertices $P_1,\ldots,P_4$ as the points $z_1, \ldots, z_4$ on the unit circle.

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