Angela the amoeba, in contemplating the meaning of her life, decided that it was to find a shape for which no regular pentagon exisited whose corners touched her outline. For example, this shape didn't work...

...because a regular pentagon exists whose corners all touch her outline:

Can Angela the Amoeba satisfy her meaning of life?

Extensions:

After attaining her meaning of life, Angela the Amoeba decided she needed something new and far more challenging. She decided to find a shape for which no equilateral triangle could be found whose vertices all lay on her outline. Prove that no matter how hard Angela tries, she can never find such a shape.

Angela eventually gave up with the triangle and started to pursue an even higher meaning of life. This was to find a shape for which no square existed whose corners touched her outline.

...because a square exists whose corners all touch her outline:

Can Angela the Amoeba satisfy her higher meaning of life? Warning: This is an unsolved problem in mathematics.

Credits:

The Extension is a classic problem.

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