How to find a line that divides two triangles in half (area) at the same time?

Question

How to find a line that divides two arbitrary triangles (in $R^2$ space) in half (area) at the same time?

We want to find a line that cut through the two triangles at the same time and each triangle is divided into two parts with equal areas.

There is a theorem call Ham Sandwich theorem (in two dimension, also called pancake theorem) saying that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. The theorems give the existence of the line.

I have no clue about how to find such a line even for two arbitrary triangles case. I am thinking about the line going through the center of each individual triangle should be the solution. However, one can easily prove that not every line going through the center (centroid) will divide the triangle in equal area. It can will prove by contradiction and using the continuity of area as the line rotates around centroid .

First, define the problem. What does it mean "divide in half"? What if triangles overlap? Then, tell us what you think. — Vasili, Mar 24 '21 at 13:26
@Vasya I think it means area. We want to show that there exists a line that cut through the two triangles at the same time, such that both parts have the same area. — 光復香港時代革命 Free Hong Kong, Mar 24 '21 at 13:30
You also need to tell people here what have you tried. Otherwise, people here won't be able to help. — 光復香港時代革命 Free Hong Kong, Mar 24 '21 at 13:38
I believe that the line through the centroids of the triangle is the answer. — Moti, Mar 24 '21 at 21:07
@Moti no, not in general. Although if I were iterating towards a solution I'd start with that. Any of the three medians through the centroid divides the triangle's area in half. This is not true for other lines through the centroid — Joffan, Mar 24 '21 at 23:58
In second thought, the probability is quite low that you will find such line - that will cut both triangles at the same time into half - there are two infinite sets of lines that cut each triangle into two equal area halves and you need to find a line that belongs to both sets. — Moti, Mar 25 '21 at 02:12
Is it necessary that hyperplane should be a line? Can't it be another curve e.g. a circle? — cosmo5, Mar 25 '21 at 06:57
@cosmo5 I believe the definition of hyperplane in 2D dimension is just a line. — Lala James, Mar 26 '21 at 23:48

score 3 · Answer 1 · answered Apr 01 '21 at 17:05

Every tangent to one of the black curves cuts the area of the triangle in half:

The curves are hyperbolae each determined by two medians which are tangent to them in the vertices of the triangle and another tangent shown in grey which is parallel to one side of the triangle. These are obtained in the following way: $\frac{bh}{2} = ax \quad \frac{a}{x} = \frac{b}{h} \; \to \frac{h^2}{2} = x^2 \quad x = \frac{h}{\sqrt{2}}$ .

For two triangles you have to find the common tangent of the hyperbolae.

How to find a line that divides two triangles in half (area) at the same time?

1 Answers1