A Porism Concerning Cyclic Quadrilaterals

We present a geometric theorem on a porism about cyclic quadrilaterals, namely, the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as for the circle. 1. Introduction Inscribe a butterfly-like quadrilateral in a circle and draw a line , see Figure 1. The sides of the quadrilateral will cut the line at four (not necessarily distinct) points. It turns out that as we continuously deform the inscribed quadrilateral, the points of intersection remain invariant. Figure 1: Porism on concyclic quadrilaterals through collinear points. More precisely, think of the quadrilateral as a path from vertex through collinear points , , , and back to (see Figure 4, left). If we redraw the path starting from another point on the circle but passing through the same points on the line in the same order, the path closes to form an inscribed polygon; that is, we will arrive at the starting point. In spirit, this startling property is similar to Steiner’s famous porism [1, 2], which states that once we find two circles, one inner to the other, such that a closed chain of neighbor-wise tangent circles inscribed in the region between them is possible, then an infinite number of such inscribed chains exist (Figure 2). One may set the initial circle at any position and the chain will close with tangency. Yet, another geometric phenomenon in the same category is Poncelet’s porism [3–5]. Figure 2: Steiner’s porism. The chain may be redrawn starting from arbitrary position of the starting circle. The property for the cyclic quadrilateral described at the outset may be restated similarly: if four points on a line admit a cyclic quadrilateral, then an infinite number of such quadrilaterals inscribed in the same circle exist. Hence, the term porism is justified. In the next sections, we restate the theorem and define a map of reversion through point, which may be represented by pseudounitary matrices. The technique developed allows one to prove the theorem as well as a diagrammatic representation of the relativistic addition of velocities, presented elsewhere [6]. A slight modification to arbitrary two-dimensional Clifford algebras allows one to modify the result to hold for hyperbolas and provides a geometric realization of trigonometric tangent-like addition. 2. The Main Result Let us present the result more formally. Reversion of a point on a circle through point