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The kindest (shortest) cut of all



Here is a piece of plywood in the shape of an isosceles triangle.

The side lengths are 1, 1, sqrt(2) units.

Quick and dirty representation:


| \

| \

| \

| \


The angle at B is a right angle.

We'd like to cut this into two pieces of equal area.

There are many ways to do this with a single cut.

Which cut has the shortest distance?

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@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest? Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter. The possibilities are endless.

Yes, it's called the isoperimetric problem, generally stated,

what is the curve of constant length that encloses the greatest area?

Equivalently, what shape of constant area has the smallest perimeter?

The answer is a circle. A 45o arc can be reflected 7 times into a full circle.

Symmetry demands this answer, although rigorous proofs abound.

Here is a general discussion.

In three dimensions (think soap bubble) a sphere encloses more volume

than any other closed surface of the same area.

isoperimetric = iso (same) perimetric (perimeter).

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