TITLE: Area method in Lobachevskian geometry
QUESTION [8 upvotes]: There are many proofs in Euclidean geometry using the area method; for example, Ceva's theorem or the proof of Pythagorean theorem shown below.


Do you know such proofs in hyperbolic geometry? 

I mean something where area helps, but does not appear in the statement.
I suspect that the answer is no, but I hope to be surprised.

REPLY [2 votes]: Well, you may get hyperbolic Ceva's theorem using area in the manner similar to Euclidean's one. Namely, using the formula for the area $E$ of a triangle with sides $a,b,c$ and angles $A,B,C$: $\sin \frac{E}2 \cosh \frac{c}2=\sinh \frac{a}2\sinh\frac{b}2\sin C$ (analogue of Euclidean $E=\frac12ab\sin C$) we get the following interpretation of cevian: a line $\ell$ through a vertex $A$ of triangle $\triangle ABC$ is a locus of points $P$ for which the ratio
$$\sin \frac{E(PAB)}2\cdot\cosh\frac{BP}2:\sin \frac{E(PAC)}2\cdot\cosh\frac{PC}2$$
is a constant which I denote $\varkappa(\ell;BA,CA)$. The above areas should be oriented but let's ignore this. When are three lines $\ell_a$ through $A$, $\ell_b$ through $B$, $\ell_c$ through $C$ concurrent? 
When $\varkappa(\ell_a;BA,CA)\cdot \varkappa(\ell_b;CB,AB)\cdot \varkappa(\ell_c;AC,BC)=1$. 
(Again, I ignore the question of concurrent lines without common point which should be treated differently in hyperbolic geometry.) 
Now if $\ell_a\cap BC=A_1$, we have 
$$
\varkappa(\ell_a;BA,CA)=
\sin \frac{E(A_1AB)}2\cdot\cosh\frac{BA_1}2:\sin \frac{E(A_1AC)}2\cdot\cosh\frac{CA_1}2=\\
\frac{\sinh\frac{BA_1}2\cosh\frac{BA_1}2}{\cosh \frac c2}:
\frac{\sinh\frac{CA_1}2\cosh\frac{CA_1}2}{\cosh \frac b2}=
\frac{\sinh BA_1}{\cosh \frac c2}:
\frac{\sinh CA_1}{\cosh \frac b2}
$$
that yields hyperbolic Ceva.