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, ..., P_n (n > 1) on the circle (O) with diameter d. s₁, s₂, ..., s_n are lengths of segments PP₁, PP₂, ..., PP_n respectively while d₁, d₂, ..., d_n are distances from P to segments P₁P₂, P₂P₃, ..., P_nP₁ respectively. Then

(For n=3, this has been shown separately.) In general, we thus have

Theorem 1

If P, P₁, P₂, ... P_n (n > 1) are concyclic on a circle with diameter d then

where, for i = 1, ..., n, s_i is the length of the segment from PP_i, d_i is the distance from P to the side P_{i i+1}, where indices are counted cyclically 1, ..., n, n+1 = 1.

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, ..., PP_n 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

Theorem 2

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 n = 4.

Consider the following three polygons with vertices P₁, P₂, P₃, P₄:

1. P₁P₂P₃P₄

   • sides: P₁P₂, P₂P₃, P₃P₄, P₄P₁
   • distances from P to these sides: d₁₂, d₂₃, d₃₄, d₄₁.
   
   

2. P₁P₂P₄P₃

   • sides: P₁P₂, P₂P₄, P₄P₃, P₃P₁
   • distances from P to these sides: d₁2, d₂4, d₄₃, d₃₁
   
   

3. P₂P₃P₁P₄

   • sides: P₂P₃, P₃P₁, P₁P₄, P₄P₂
   • distances from P to these sides: d₂₃, d₃₁, d₁₄, d₄₂.

Obviously in our notations d_{i j} = d_{j i}.

By our theorems,

From these we can have following other equal products of distances:

This result can be formulated as

Theorem 3

Under the assumptions of Theorem 1, for n = 4, the products of the distances from a point on the circumcircle of a cyclic quadrilateral to a pair of opposite sides is the same as for the other pair and for the diagonals.

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