If you know Brouwer's fixed point theorem in the plane and do not consider that to be complicated, then no. The curious can DM me and I will share a PDF of this little article (it is three pages).
If you know some basic Algebraic Topology (homology), Hatcher gives a proof in section 2.B for the theorem (actually, he proves something even stronger) in a little under a page.
Don't we usual define shapes with points? Like the corners of a triangle, since they have defined coordinates. Would a point at the same coordinate be inside or out?