Proof of the Hyperbolic Triangle Theorem

The Hyperbolic Triangle Theorem
The sum of angles of a hyperbolic triangle is less than π.

Let PQR be an h-triangle.
By the Origin Lemma, there is an h-inversion i mapping P to O.
Let Q' = i(Q) and R' = i(R).
Since iεH(2), the angles of the h-triangle OQ'R' are equal to those of PQR.

The h-segment Q'R' does not pass through O, so is an arc of a circle K.
As K is orthogonal to C, O lies outside K, so that
the h-triangle OQ'R' lies inside the euclidean triangle OQ'R'.
This is shown in the CabriJava figure.

The tangents to K at Q' and R' give the angles of the h-triangle,
so these are less than the angles of the euclidean triangle at Q'and R'.
The angle at O is the same for both.

The result follows since the sum of angles of the euclidean triangle is π.

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