On Kakeya's Geometric Proof of Eneström-Kakeya's Theorem

Abstract

This paper is devoted to demonstrate Kakeya's geometric proof of his theorem (1912), independently established earlier by Eneström (1893). By calculating centers and radii of the interlacing circles of Kakeya's method, we prove Kakeya's geometric structure, which has not been previously established. We give an equivalent proof, which is based on the construction of internally interlacing circles, which has been geometrically considered by Tomic (1948).

Figures (9)

• Figure 1: (a) $\triangle OC_0R_0\cong\triangle R_0C_0S_1$, since $OR_0=R_0S_1$. (b) $R_0$, $C_0$ and $C_1$ are collinear as Eq. (\ref{['eq:2.4']}) confirms.
• Figure 2: The construction of $\mathcal{C}_0,\,\mathcal{C}_1,\,\mathcal{C}_2$ when $\theta=\frac{\pi}{3}$. (a) $p_0=1, p_1=2, p_2=3$ are different. (b) $p_0=1, p_1=p_2=2$ and $\mathcal{C}_1,\,\mathcal{C}_2$ are identical. Accidentally $C_1$ lies on $\mathcal{C}_0.$ Notice that $C_2$ does not lie on $\mathcal{C}_1$.
• Figure 3: The circles $\mathcal{C}_0,\,\mathcal{C}_1,\,\mathcal{C}_2$ with a right angle $\theta=\frac{\pi}{2}$. (a) $p_0=1, p_1=2, p_2=3$ are different. (b) $p_0=1, p_1=p_2=2$ and $\mathcal{C}_1,\,\mathcal{C}_2$ are identical.
• Figure 4: The construction of $\mathcal{C}_0,\,\mathcal{C}_1,\,\mathcal{C}_2$ when $\theta=\frac{2\pi}{3}$. (a) $p_0=1, p_1=2, p_2=3$ are different. Accidentally $C_2$ coincides with the origin. (b) $p_0=1, p_1=p_2=2$ and $\mathcal{C}_1,\,\mathcal{C}_2$ are identical.
• Figure 5: The construction of $\mathcal{C}_k,\,\mathcal{C}_{k+1},\,\mathcal{C}_2$. (a) $C_{k+1}$ lies entirely outside $C_k$ and it only touches it at $R_k$. (b) $C_{k+1}$ lies entirely outside $C_k$ and it only touches it at $R_k$ when$\theta^*$ lies in the third or fourth quadrants.

Theorems & Definitions (4)

• Lemma 1.1
• Remark 3.1
• Lemma 3.2
• proof