A pentagon in a pentagon in a pentagon

[BMAD]

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?

[bonanova]

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

True?

[BMAD]

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

True?

yes. any pentagon

[bonanova]

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

[BMAD]

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

I have only started analyzing this.

Not all them.

[bonanova]

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?

[BMAD]

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.

[BMAD]

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