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Limiting Behavior

We have seen that the total inradii of regular polygons tend to the diameter of the circle. We now prove a slightly different theorem on the limiting behavior of cyclic polygons.

**Theorem**. *Let \(A = \{ a_k : k \in {\mathbb Z}^+ , a_i \neq a_j \) for \( i \neq j\} \) ** be a dense subset of a circle of radius \(R\) and let *\(P(n)\) *be the convex polygon formed by the vertices \( \{a_k : 1 \leq k \leq n \} . \) Then the sequence *\( \left( r_{P(n)} \right)_{n=3}^{\infty} \) *is eventually increasing and *\( \lim_{n \rightarrow \infty} r_{P(n)} = 2R . \)

We leave the proof of the following lemma to the reader.

**Lemma**. \( \cos (x) + \cos (y) \geq \cos(x+y) + 1 \) for any \( (x,y) \) *in the region bounded by \( x = 0, y = 0, \) and \( x + y = \pi , \) with equality only on the boundary*.

**Proof of theorem.** Because \(A\) is dense, there is an \(M\) such that the angle measures for every side of \(P(M)\) is less than \( \pi / 4 .\) Without loss of generality we may represent \(P(n+1) \) and \(P(n) \) as \( (\theta_1, \ldots, \theta_{n+1}) \) and \( ( \theta_1, \ldots, \theta_{n-1}, \theta_n + \theta_{n+1}) ,\) respectively. So,

\[ r_{P(n+1)} = R \left( 2 - (n+1) + \sum_{k=1}^{n+1} \cos \left( \frac{\theta_k}{2} \right) \right) \]

and

\[ r_{P(n)} = R \left( 2 - n + \sum_{k=1}^{n-1} \cos \left( \frac{\theta_k}{2} \right) + \cos \left( \frac{\theta_n + \theta_{n+1}}{2} \right) \right) . \]

From the preceding lemma it follows that \( r_{P(n)} < r_{P(n+1)} . \) So the sequence \( \left( r_{P(n)} \right)_{n=3}^{\infty} \) is eventually increasing.

By our theorem on regular polgyons we know that for a given number of vertices, the largest radial sum is obtained from the regular \(n\)-gon and this value is bounded above by \(2R .\) Consequently, the sequence \( \left( r_{P(n)} \right) \) is eventually increasing and is bounded above by \( 2R .\) Thus, to prove the theorem it suffices to show that for a given \( \epsilon > 0 , \) there exists an \(N\) such that \( r_{P(N)} > 2R - \epsilon .\) By the limiting theorem for regular polygons there exists an \( m > 4 \) such that \( r_{P_m} > 2R - \epsilon / 2 , \) where \( P_m \) is a regular \(m\)-gon. Since \(A\) is a dense set, there exist \(m\) distinct points \(A^{\prime} = \{ a_{n_1}, a_{n_2}, \ldots, a_{n_m} \} \subset A\) close enough to the \(m\) vertices of \( P_m \) that the polygon \( P^{\prime} \) with vertex set \( A^{\prime} \) has \( r_{P^{\prime}} > 2R - \epsilon \).∎