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# Polygons

## Question

Let's define BMAD polygons as polygons where the perimeter and area values are identical. Does there exist two such polygons (p1, p2) where the number of sides of p1 > the number of sides of p2 but the perimeters are identical?

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Let p2 be equilateral triangle with side of length 4*3^0.5. Perimeter and area of p2 are both equal to 12*3^0.5.

Let p1 be rhombus with side of length 3*3^0.5. P1 and p2 have equal perimeters.
Obviously we can make p1's area anything between 0 (when shorter diagonal of p1 tends to 0) and 27 (when p1 becomes a square)
without changing it's perimeter, so we can make it equal to 12*3^0.5 (=~20.785) in particular.

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Regular?

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one such polygon exists where perimeter = area, then put an arbitrary point O on any one of its sides AB. introduce a infinitesimally small kink at O, such that AO and OB are almost almost almost colinear but are not.

this new polygon has the same area and same perimeter as the old one so it satisfies the original conditions and also has greater number of sides.
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Regular?

no not necessarily.

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one such polygon exists where perimeter = area, then put an arbitrary point O on any one of its sides AB. introduce a infinitesimally small kink at O, such that AO and OB are almost almost almost colinear but are not.

this new polygon has the same area and same perimeter as the old one so it satisfies the original conditions and also has greater number of sides.

wouldn't the area be infinitesimally off

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For example:

Perimeter = 4 units

Area = 4 units^2

(assuming the above is possible)

Could I find a BMAD Triangle where

Perimeter = 4 units

Area = 4 units^2

assuming the same size unit in both cases

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Does this hold for shapes with sides > 3? >4?

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one such polygon exists where perimeter = area, then put an arbitrary point O on any one of its sides AB. introduce a infinitesimally small kink at O, such that AO and OB are almost almost almost colinear but are not.

this new polygon has the same area and same perimeter as the old one so it satisfies the original conditions and also has greater number of sides.

wouldn't the area be infinitesimally off

yup, sowwee

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nice.

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