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.