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A pentagon in a pentagon in a pentagon

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Conjecture: As long as each of the two smaller pentagons are formed by joining the midpoints of the next larger pentagon, the two ratios of the areas will always be equal and the two ratios of the perimeters will always be equal, no matter what shape the largest pentagon has.

Under what conditions is this true?

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Posted · Report post

Conjecture: By saying "whatever shape the pentagon has" you are not limiting the discussion to regular pentagons.

True?

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Posted · Report post

Conjecture: By saying "whatever shape the pentagon has" you are not limiting the discussion to regular pentagons.

True?

yes. any pentagon

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Posted · Report post

If the pentagons are convex it looks like a sure bet.
I have only started analyzing this.

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Posted · Report post

If the pentagons are convex it looks like a sure bet.

I have only started analyzing this.

Not all them.

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Posted · Report post

If the pentagons are convex it looks like a sure bet.

I have only started analyzing this.

Not all them.

I'm not making any progress. Any tips for analyzing this?

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Posted · Report post

I've only been able to find very specific cases, creating perpendicular line segments from midpoints of the innermost pentagon to the original pentagon but don't understand why this works or if this is the only case where it works.

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Posted · Report post

I've only been able to find very specific cases, creating perpendicular line segments from midpoints of the innermost pentagon to the original pentagon but don't understand why this works or if this is the only case where it works.

So far all of the the ones that fit this conjecture all fit the golden ratio

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