A consistency equation for three geometries (Q1065390)

Dubikajtis (oral communication of the author) has recently introduced the so-called S-geometry. Two axioms are adopted in the paper and an orthogonality function defined. ''Some elementary but long and tedious calculations'' (not reproduced here) have led Dubikajtis to the functional equation \[ (*)\quad \frac{f(x)+f(y)}{f(x+y)}+\frac{f(x)-f(y)}{f(x-y)}=2. \] The main part of the paper deals with a proof of the theorem that, if f satisfies certain conditions, the equation (*) has three kinds of solutions. A conclusion reads that any theorem in S-geometry yields a common property of Euclidean, hyperbolic and elliptic geometry.