Question 53
Simple polygon $A$ with perimeter $p$ is called a **rotund polygon**, if for each two points $x$ and $y$ on the perimeter of $A$ that have a distance of at most $1$ in the plane, their distance on $A$ (i.e. the smaller part of the perimeter of $A$ that lies between them) is at most $\frac{p}{4}$. We want to prove that a circle with radius $\frac{1}{4}$ can be drawn entirely inside a rotund polygon.
Intellectuals of the Earth and researchers of the planet **Hot Dog**! have devised two completely different approaches to solve the problem. In both approaches a **chord** is a line segment with its end points lying on the perimeter of the polygon. A **diameter** is a chord with its endpoints being vertices of the polygon. An **inner chord** is a chord that lies entirely inside or on the perimeter of the polygon. The **distance on the perimeter** between two points on the polygon is defined as the length of the smaller part of the perimeter between them.
### The Earth approach: the maximum chord
We know as a fact that for each polygon, an inner chord $xy$ with a length less than or equal to unity can be found such that for each inner chord $x'y'$ with a length less than or equal to unity, the distance on the perimeter of $x$ and $y$ is greater than or equal to the distance on the perimeter of $x'$ and $y'$. This chord is called **the maximum chord**. In a rotund polygon $A_o$ there are two possibilities for the maximum chord:
a) **First possibility:** The length of the maximum chord equals unity. Prove that in this case a semicircle with the maximum chord as its diameter can fit completely inside the polygon $A_o$; therefore, a circle with radius $\frac{1}{4}$ can be drawn entirely inside the polygon.
b) **Second possibility:** The length of the maximum chord is less than unity. Prove that still in this case one can find a circle with radius $\frac{1}{4}$ that fits completely inside the polygon $A_o$.
Intellectuals of the Earth thought for many times that they have tackled the problem, but each time they noticed a small flaw in their proof, until they could finally solve it.
### The Hot Dog approach: triangulation
Consider the two statements below:
**First statement:** "For each arbitrary polygon with edges of at most unity length for which there are no circles of radius $1/4$ that could be fit entirely inside it, it's possible to triangulate it with diameters of at most unit length."
**Second statement:** "For each arbitrary polygon that a circle of radius $1/4$ cannot be fit entirely inside it, it's possible to triangulate it with chords of at most unit length."
Researchers of Hot Dog could prove that if the second statement is true, then one can draw a circle of radius $1/4$ completely inside a rotund polygon.
c) Prove that if the second statement is true, then one can draw a circle of radius $1/4$ completely inside a rotund polygon.
They could easily deduce that if the first statement is true, then the second one is also true. So they announced that a house full of Hot Dogs would be the reward of anyone who could prove the first statement! After a while, a young barber called J.N who considered himself to be from the Earth, succeeded in contradicting the first statement and was rewarded the house, as promised.
d) Construct a 1392-gon that contradicts the first statement.
Still, researchers of Hot Dog have the hope to prove the second statement directly.
e) Write anything that you think might be true about the second statement.
Intellectuals of the Earth and researchers of the planet **Hot Dog**! have devised two completely different approaches to solve the problem. In both approaches a **chord** is a line segment with its end points lying on the perimeter of the polygon. A **diameter** is a chord with its endpoints being vertices of the polygon. An **inner chord** is a chord that lies entirely inside or on the perimeter of the polygon. The **distance on the perimeter** between two points on the polygon is defined as the length of the smaller part of the perimeter between them.
### The Earth approach: the maximum chord
We know as a fact that for each polygon, an inner chord $xy$ with a length less than or equal to unity can be found such that for each inner chord $x'y'$ with a length less than or equal to unity, the distance on the perimeter of $x$ and $y$ is greater than or equal to the distance on the perimeter of $x'$ and $y'$. This chord is called **the maximum chord**. In a rotund polygon $A_o$ there are two possibilities for the maximum chord:
a) **First possibility:** The length of the maximum chord equals unity. Prove that in this case a semicircle with the maximum chord as its diameter can fit completely inside the polygon $A_o$; therefore, a circle with radius $\frac{1}{4}$ can be drawn entirely inside the polygon.
b) **Second possibility:** The length of the maximum chord is less than unity. Prove that still in this case one can find a circle with radius $\frac{1}{4}$ that fits completely inside the polygon $A_o$.
Intellectuals of the Earth thought for many times that they have tackled the problem, but each time they noticed a small flaw in their proof, until they could finally solve it.
### The Hot Dog approach: triangulation
Consider the two statements below:
**First statement:** "For each arbitrary polygon with edges of at most unity length for which there are no circles of radius $1/4$ that could be fit entirely inside it, it's possible to triangulate it with diameters of at most unit length."
**Second statement:** "For each arbitrary polygon that a circle of radius $1/4$ cannot be fit entirely inside it, it's possible to triangulate it with chords of at most unit length."
Researchers of Hot Dog could prove that if the second statement is true, then one can draw a circle of radius $1/4$ completely inside a rotund polygon.
c) Prove that if the second statement is true, then one can draw a circle of radius $1/4$ completely inside a rotund polygon.
They could easily deduce that if the first statement is true, then the second one is also true. So they announced that a house full of Hot Dogs would be the reward of anyone who could prove the first statement! After a while, a young barber called J.N who considered himself to be from the Earth, succeeded in contradicting the first statement and was rewarded the house, as promised.
d) Construct a 1392-gon that contradicts the first statement.
Still, researchers of Hot Dog have the hope to prove the second statement directly.
e) Write anything that you think might be true about the second statement.