In the proof of Cauchy-Goursat Theorem, the following fact was used:
The distance between any point $z$ on a triangle and a point $z_0$ interior to the triangle is less than half the perimeter of the triangle.
I think it is true, but I don't know how to prove it. I'd like to prove the above assertion rigorously.
Please let me know if you have any comment for this. Thanks in advance.
Consider the line through $z$ and $z_0$. It meets the triangle again at $z_1$. Then $d(z,z_0)<d(z_1,z_0)$. But $z_0$ and $z_1$ divide the boundary of the triangle into two paths from $z_0$ to $z_1$. One of these has length $l$ which is at most half the perimeter. But by the triangle inequality, $d(z_1,z_0)\le l$.