geometry - Nice proof of the triangle inequality for the metric of the hyperbolic plane

We need to check $eta(u,v)+eta(v,w)geeta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$.

First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an isometry, we may now assume that $v=(0,0,1)$. This is the main idea. For added convenience, you may also rotate the $xy$-plane so that the $y$-coordinate of $u$ equals 0.

Next, observe that the formula yields equality in the case when the projections of $u$ and $w$ to the $xy$-plane are endpoints of a segment containing (0,0). This is straigtforward if you write $u=(sinh a,0,cosh a)$ and $w=(-sinh b,0,cosh b)$ where $a,bge 0$.

Finally, rotate $w$ around the $z$-axis until it comes to a position as above. The product $ucdot w$ grows down (it equals contant plus the scalar product of the $xy$-parts, since $z$-coordinates are fixed). Hence $eta(u,w)$ grows up while the two other distances stay, q.e.d.

Of course, for writing purposes the last step is just an application of Cauchy-Schwartz for the scalar product in $mathbb R^2$.

This was about two-dimensional hyperbolic plane, in higher dimensions just insert more coordinates (they will not actually show up in formulae).