Relations in a Cyclic Polygon
The results below are due to Bui Quang Tuan, a contemporary Vietnamese mathematician.
Starting with one of his theorems concerning two cyclic n-gons with a common circumcircle we establish a special case where one of the polygons shrinks into a point.
Given point P and n points P1, P2, ..., Pn
(For n=3, this has been shown separately.) In general, we thus have
If P, P1, P2, ... Pn
|(1)||(s1·s2· ... ·sn)2 = d1·d2· ... ·dn·dn|
Adding some verbiage, (1) reads:
Of course, there are many polygons with the same set of vertices, and the distances from a point to the sides of a polygon vary from one polygon to another. However, obviously, all such polygons share the same set of segments PP1, PP2, ..., PPn and hence the product of such segments from P to the n vertices in the left-hand side of (1) does not depend on the selection of the polygon. This leads to
Under the assumptions of Theorem 1, the product of all distances from a point to the sides of a polygon with a given set of vertices is independent of the polygon:
Of course some polygons will share some sides as well, not only the vertices. We look into a special example with
Consider the following three polygons with vertices P1, P2, P3, P4:
- sides: P1P2, P2P3, P3P4, P4P1
- distances from P to these sides: d12, d23, d34, d41.
- sides: P1P2, P2P4, P4P3, P3P1
- distances from P to these sides: d12, d24, d43, d31
- sides: P2P3, P3P1, P1P4, P4P2
- distances from P to these sides: d23, d31, d14, d42.
Obviously in our notations
By our theorems,
From these we can have following other equal products of distances:
This result can be formulated as
Under the assumptions of Theorem 1, for