Abstract
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.
Highlights
Inscribe a butterfly-like quadrilateral in a circle and draw a line L, see Figure 1
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
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, an infinite number of such inscribed chains exist (Figure 2)
Summary
Inscribe a butterfly-like quadrilateral in a circle and draw a line L, see Figure 1. 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, an infinite number of such inscribed chains exist (Figure 2). 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
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