Say we have a triangle $ABC$. I want to find a curve $\gamma:[0,1]\to\mathbb{R}^2$ such that $\gamma(0)=A$, $\gamma(1)=B$ and for all $t\in(0,1)$ the tangent line at $\gamma(t)$ divides $\triangle ABC$ into two pieces of same area (a smaller triangle and a quadrilateral). The curve can be as smooth as you might need of course.
I came up with this problem about a week ago and got a bunch of few equations at first. Now I'm otherwise engaged so I'm posting the problem before I forget about it. Cheers!