I'm trying to solve this problem.
Let $E=\mathbb{R}^{2}$ with the Euclidian metric $d_{2}$. Fix a point $P \in E$ and define a map $d : E \times E \to \mathbb{R}$ by $$d(x, y) = \begin{cases} d_{2}(x, y) & \text{if } x, y, P \text{ are on the same line } \\ d_{2}(x, P) + d_{2}(y, P) & \text{otherwise} \end{cases}$$
Prove that $d$ is a metric on $E$.
I found that to prove the triangular inequality by case-by-case ($x,y,z$ are on the same line, v.v..) is rather boring.
I would like to ask if I can prove $d(x,y) \le d(x,z) + d(y,z)$ more directly. Please shed me some light!
My attempt:
We have $$d(x,z) \ge \min\{d_{2}(x, z), d_{2}(x, P) + d_{2}(z, P)\}$$ $$d(y,z) \ge \min\{d_{2}(y, z), d_{2}(y, P) + d_{2}(z, P)\}$$ $$d(x,y) \le \max\{d_{2}(x, y), d_{2}(x, P) + d_{2}(y, P)\}$$
I fail to manipulate to get the desire inequality.