Diagram of regular polygon and exterior point

[Trenters4325]

Point B is in the exterior of the regular n-sided polygon $\displaystyle A_1A_2\cdots A_n, and A_1A_2B$is an equilateral triangle. What is the largest value of n for which $\displaystyle A_1, A_n,$and B are consecutive vertices of a regular polygon?

Can someone draw a diagram of this for me? I just can't visualize it.

 

[pka]

Point B is in the exterior of the regular n-sided polygon $\displaystyle A_1A_2\cdots A_n, and A_1A_2B$is an equilateral triangle. What is the largest value of n for which $\displaystyle A_1, A_n,$and B are consecutive vertices of a regular polygon? I just can't visualize it.

I bet can’t either! Not sure anyone can.

If that description is correct, then $\displaystyle m\left( {\angle A_1 BA_2 } \right) = \frac{\pi }{3}$!

But in order $\displaystyle A_1, A_n,$and B are consecutive vertices of a regular polygon to be true, the angle $\displaystyle {\angle A_1 BA_2 }$ would be one of the polygon’s interior angles. Now that cannot happen unless n=2, but that makes no sense.

This makes it impossible: consecutive vertices of a regular polygon.

 

[soroban]

Hello, Trenters4325!

Point $\displaystyle B$ is in the exterior of the regular $\displaystyle n$-sided polygon $\displaystyle A_1,\,A_2,\,\cdots\, A_n$

$\displaystyle \;\;$and $\displaystyle A_1A_2B$is an equilateral triangle.

What is the largest value of $\displaystyle n$ for which $\displaystyle A_1,\,A_n,\,$and $\displaystyle \,B$ are consecutive vertices of a regular polygon?

                   B
                   o
                  / \
                 /   \
                /     \
               /60°    \
           A1 o * * * * o A2

             *           *

            *             *

        An o               o A3
           *               *
           *               *
           *               *

Hope that's enogh to get you started . . .

 

[soroban]

Hello, Trenters4325!

The way I interpreted the problem, it is possible . . .
I think they meant: $\displaystyle \,A_1,\;A_n,\;B$ in some order are consecutive vertices.

                   B
                   o
                  / \
                 /   \
                /     \
               /60°    \
           A1 o * * * * o A2

             *           *

            *             *

        An o               o A3
           *               *
           *               *
           *               *

For a regular $\displaystyle n$-gon, an interior angle, say, $\displaystyle \angle A_nA_1A_2.\,$ is equal to: $\displaystyle \,\frac{180(n\,-\,2)}{n}$ degrees.
Hence, $\displaystyle \angle A_nA_1B \:=\:\frac{180(n\,-\,2)}{n}\,+\,60$ degrees.

This angle is to be an interior angle of another regular polygon of, say, $\displaystyle k$ sides.
A regular $\displaystyle k$-gon has an interior angle of: $\displaystyle \,\frac{180(k\,-\,2)}{k}$ degrees.

So we have: $\displaystyle \,\frac{180(n\,-\,2)}{n}\,+\,60\;=\;\frac{180(k\,-\,2)}{k}$

Solve for $\displaystyle k:\;\;k \:=\:\frac{6n}{6\,-\,n}$

Since $\displaystyle k$ is a positive integer, there are but a few solutions:
$\displaystyle \;\;(n,\,k)\;=\;\sout{(2,3)},\;(3,6),\;(4,12),\;(5,30)$

The largest value is: $\displaystyle \,n\,=\,5$


If we start with a regular pentagon,
the new interior angle $\displaystyle (108\,+\,60\:=\:168^o)$ belongs to a regular 30-gon.