Some new theorems on Pentagon and Pentagram

TL;DR

This paper presents new theorems related to pentagons and pentagrams, expanding the mathematical understanding of these geometric figures.

Contribution

It introduces novel theorems on pentagon and pentagram geometry, contributing new insights to classical geometric studies.

Findings

New theorems on pentagon properties

New theorems on pentagram properties

Enhanced understanding of geometric relationships

Abstract

We establish some new theorems on pentagon and pentagram.

Full text

1. Recall three classical theorems

Miquel’s Pentagram Theorem can be considered as one of the most beautiful theorem on pentagram.

Theorem 1** (Miquel’s Pentagram Theorem [1]).**

Consider a convex pentagon and extend the sides to a pentagram. Externally to the pentagon, there are five triangles. Construct the five circumcircles. Each pair of adjacent circles intersects at a vertex of the pentagon and a second point. Then Miquel’s pentagram theorem states that these five second points are concyclic.

When five circumcenters are concyclic, we get the particular case of Miquel’s Pentagram Theorem that is ”Miquel Five Circles Theorem” with ten cyclic points.

Theorem 2** (Miquel Five Circles Theorem [2]).**

Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. By joining adjacent pairs of the intersection points which do not lie on the circle of center, an (irregular) pentagram is obtained each of whose five vertices lies on one of the circles with concyclic centers.

The following theorem was discovered in 1989 by high school student Takada [3]. This theorem is very beautiful and is also considered one of the classic theorems on the pentagon. Takada’s theorem below is very close to Miquel’s Pentagram theorem.

Theorem 3** (Takada’s theorem [3]).**

Consider a cyclic pentagon. Circumcircles of the triangles whose vertexs are the intersections of each pair of its diagonals and the vertex of pentagon. Each pair of adjacent circles intersect at a vertex of the pentagon and a second point. Then Takada’s theorem states that these five second points are also concyclic.

2. Main theorems

In this section, we shall use the above classic theorems to create some interesting problems with cyclicity and collinearity on the pentagon and pentagram.

Theorem 4** (Tran Quang Hung’s Elevent Circles Theorem).**

Let $A_{i}$, $i=1,2,\ldots,5$, be any five points. Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $A_{i}A_{i+1}$ and $A_{i+2}A_{i+3}$ by $B_{i+3}$, the second intersection of two circles $(A_{i}A_{i+1}B_{i+2})$ and $(A_{i+1}A_{i+2}B_{i+3})$ by $C_{i+1}$, the center of circle $(A_{i}A_{i+1}B_{i+2})$ by $K_{i+2}$, and the center of circle $(C_{i+1}B_{i+2}B_{i+3})$ by $L_{i}$. Then five lines $K_{i}L_{i}$, for $i=1,2,\ldots,5$, are concurrent at a point $X$.

Theorem 5** (Dual of Theorem 4).**

Let $A_{i}$, $i=1,2,\ldots,5$, be any five points. Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $A_{i}A_{i+1}$ and $A_{i+2}A_{i+3}$ by $B_{i+3}$, the second intersection of two circles $(A_{i}A_{i+1}B_{i+2})$ and $(A_{i+1}A_{i+2}B_{i+3})$ by $C_{i+1}$, and the center of circle $(A_{i}C_{i+1}A_{i+2})$ by $L_{i}$. Follow Miquel’s theorem then five points $B_{i+1}$, $C_{i}$, $A_{i+1}$, $C_{i+2}$, and $B_{i+4}$ lies on a circle $(K_{i})$ with taking subscripts modulo $5$. Then five lines $K_{i}L_{i}$, for $i=1,2,\ldots,5$, are concurrent at a point $X$.

Theorem 6** (The first theorem of collinearity with Twelve Circles).**

Let $A_{i}$, $i=1,2,\ldots,5$, be any five points. Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $A_{i}A_{i+1}$ and $A_{i+2}A_{i+3}$ by $B_{i+3}$, the second intersection of two circles $(A_{i}A_{i+1}B_{i+2})$ and $(A_{i+1}A_{i+2}B_{i+3})$ by $C_{i+1}$, the center of circle $(A_{i}A_{i+1}B_{i+2})$ by $K_{i+2}$, and the center of circle $(C_{i+1}B_{i+2}B_{i+3})$ by $L_{i}$.

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Assume that $B_{i}$, for $i=1,2,\ldots,5$, lies on the circle $(O)$.

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Follow Theorem 1, we have five points $C_{i}$, for $i=1,2,\ldots,5$, are concyclic on circle $(J)$.

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Follow Theorem 4, we have five lines $K_{i}L_{i}$, for $i=1,2,\ldots,5$, are concurrent at a point $X$.

The theorem 6 states that three points $O$, $J$, and $X$ also are collinear.

Theorem 7** (The second theorem of collinearity with Twelve Circles).**

Let $A_{i}$, $i=1,2,\ldots,5$, be any five points. Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $A_{i}A_{i+1}$ and $A_{i+2}A_{i+3}$ by $B_{i+3}$, the second intersection of two circles $(A_{i}A_{i+1}B_{i+2})$ and $(A_{i+1}A_{i+2}B_{i+3})$ by $C_{i+1}$, the center of circle $(A_{i}A_{i+1}B_{i+2})$ by $K_{i+2}$, and the center of circle $(C_{i+1}B_{i+2}B_{i+3})$ by $L_{i}$.

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Assume that $A_{i}$, for $i=1,2,\ldots,5$, lies on the circle $(O)$.

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Follow Theorem 1, we have five points $C_{i}$, for $i=1,2,\ldots,5$, are concyclic on circle $(J)$.

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Follow Theorem 4, we have five lines $K_{i}L_{i}$, for $i=1,2,\ldots,5$, are concurrent at a point $X$.

The Theorem 7 states that three points $O$, $J$, and $X$ also are collinear.

Theorem 8** (The collinearity from Miquel Five Circles Theorem).**

Let $A_{i}$, $i=1,2,\ldots,5$, be any five points. Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $A_{i}A_{i+1}$ and $A_{i+2}A_{i+3}$ by $B_{i+3}$, the second intersection of two circles $(A_{i}A_{i+1}B_{i+2})$ and $(A_{i+1}A_{i+2}B_{i+3})$ by $C_{i+1}$, the center of circle $(A_{i}A_{i+1}B_{i+2})$ by $K_{i+2}$, and the center of circle $(C_{i+1}B_{i+2}B_{i+3})$ by $L_{i}$.

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Follow Theorem 1, we have five points $C_{i}$, for $i=1,2,\ldots,5$, are concyclic on circle $(O)$.

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Assume that $K_{i}$, for $i=1,2,\ldots,5$, lies on a circle, follow Theorem 2 this circle is also $(O)$.

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Taking subscripts modulo $5$, we denote, for $i=1,2,\ldots,5$, the intersection of the lines $K_{i}K_{i+2}$ and $K_{i+1}K_{i+3}$ by $D_{i+3}$, and the second intersection of two circles $(D_{i}D_{i+1}K_{i+2})$ and $(D_{i+1}D_{i+2}K_{i+3})$ by $E_{i}$.

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Follow Theorem 1, we have five points $E_{i}$, for $i=1,2,\ldots,5$, are concyclic on circle $(J)$.

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Follow Theorem 4, we have five lines $K_{i}L_{i}$, for $i=1,2,\ldots,5$, are concurrent at a point $X$.

The Theorem 8 states that

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Three points $K_{i}$, $L_{i}$, and $E_{i}$ are collinear for $i=1,2,\ldots,5$.

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Three points $O$, $J$, and $X$ also are collinear.

Remark**.**

We used the result of Theorem 4 in the the theorems 6, 7, 8. If we use also the result of Theorem 5 (Dual of Theorem 4), we shall obatain three similar theorems like the theorems 6, 7, 8.

Acknowledgement**.**

The author is very grateful to Alexander Skutin from Russia, for his efforts in reading carefully this manuscript. The author is also very grateful to Hiroshi Okumura from Japan, for the information of Takada’s theorem.