Minimum fencing when dividing triangle into 4 parts

[jonas.hall]

So I have an equilateral triangle an I want to divide it in 4 parts, all having the same area. This can be done in a multitude of ways of course. But assuming it's a garden and the division is about putting up a fence, which division uses the least fencing?

Now I have two alternatives so far.

The first is to create a cirle in the middle and add three short segments from the circle to the midpoints of the sides. There should be a uniqe such solution and I just haven't bothered calculating it yet.

The second is, in my opinion more interesting. Cut of the corners with a curve symmetrical around the bisectors. Now if this curve was a straight line at right angles to the bisector it would be uniquely determined. Also if it was a circle centered in the vertex. After calculating some special cases one might be satisfied and pick the best but I was thinking one might set up a differential equation to find the best possible curve. But how would I set this up?

 

[gtoguy97]

Any help on how to go about this would be much appreciated. One approach to this problem is to use the calculus of variations. The goal is to minimize the length of the fencing, which can be thought of as a functional (a function of some other functions). The idea is to find a curve that minimizes the length of the fence while still dividing the triangle into four equal parts. To do this, you can use the Euler-Lagrange equations, which are derived from minimizing the functional by varying the curve. This will give you a differential equation that you can then solve for the desired curve.

 

[tacman]

There are a few different ways to approach this problem, but one possible solution would be to use a combination of straight lines and circular arcs to divide the triangle into four equal parts with the minimum amount of fencing.

First, draw a straight line from each vertex to the opposite midpoint of the opposite side. This will create three smaller triangles within the original equilateral triangle.

Next, draw a circular arc from each vertex, intersecting the two adjacent sides at their midpoints. This will create four equal-sized sections within the original triangle.

By using this method, you will only need to construct three straight lines and three circular arcs, resulting in a total of six segments of fencing. This is likely the most efficient solution in terms of minimizing the amount of fencing needed.

As for using a differential equation to find the optimal curve, it may be possible to do so but it would likely be a complex and time-consuming process. It may be more practical to use a geometric approach, as described above, to find the most efficient solution.