The problem asked
Suppose you have an equilateral triangle and you want to cut it in half using a pizza cutter (which can cut curves, not just straight lines). What is the shortest cut you can make?
Two commenters got the correct answer. The cut is one sixth of a circle whose center is at a vertex of the triangle.
It’s not immediately obvious to me why this is true. However, the following picture from Mahajan’s book makes it pretty clear.
Assuming the cut goes from one side of the triangle to another, we could tile six triangles around to make a hexagon, and find a minimum-length cut to divide all the triangles in half. As long as you believe a circle has the minimum perimeter for a fixed area, this should be pretty convincing.