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  • 4
    $\begingroup$ Could you please define "cross ratio" in this context? What specifically does $[a,b,c,d]$ mean? $\endgroup$ Commented Jul 18, 2019 at 23:24
  • 5
    $\begingroup$ Sure! $[a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}.$ $\endgroup$ Commented Jul 18, 2019 at 23:29
  • 11
    $\begingroup$ That's very cool! $\endgroup$ Commented Jul 18, 2019 at 23:52
  • 4
    $\begingroup$ I've tried, but no success. The only trivial observation I've made, is that the duality transform (in the spherical geometry) preserves a ratio (tan(a/2)/tan(b/2)), where a and b angles opposite to an edge. If we show, that your cross-ratio depends only from these six ratios, we are done. I played with nice wikipedia-formulas (see the last section), but did not managed to organize it in symmetric form. en.wikipedia.org/wiki/Spherical_trigonometry $\endgroup$ Commented Jul 21, 2019 at 18:16
  • 8
    $\begingroup$ Empirical observation: the four points $(\cot\frac{\Omega_i}2, \frac1{P_i})$ are collinear. $\endgroup$ Commented Jul 22, 2019 at 9:46