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:
A
| \
| \
| \
| \
B--------------C
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?