I'm currently reading Freedman's paper on the Mobius invariance of knot energy, and I'm stuck on a particular equality (2.8).
Let $\gamma$ be a curve in $\mathbb{R}^3$ parametrized with respect to arc length, and let $T$ be a spherical inversion which does not send the curve to infinity. Then we have that $\frac{|(T\circ \gamma)'(u)| |(T\circ\gamma)'(v)|}{|(T\circ\gamma)(u) - (T\circ\gamma)(v)|^2} = \frac{1}{|\gamma(u) - \gamma(v)|^2}$ for any $u,v$. The paper, and Wikipedia, says this is a short calculation involving the law of cosines, but I don't see how to apply it, as the tangent vectors, along with the segment joining the two points on the curves, don't form a triangle.
It turns out this is just an elementary application of the ratios of side lengths for similar triangles.
The triangle formed by $\gamma(u), \gamma(v),$ and $\gamma(u) + \gamma'(u)$ is similar to its image under spherical inversion, as the mapping is conformal. Thus, the ratios $\frac{|\gamma'(u)|}{|\gamma(u) - \gamma(v)|}$ and $\frac{|(T \circ \gamma)'(u)|}{|(T\circ\gamma)(u) - (T\circ\gamma)(v)|}$ are equal. Likewise, we have the analogous statement for the triangle formed by $\gamma(u), \gamma(v)$ and $\gamma(v) + \gamma'(v)$ and its image. Multiplying the two equalities of the ratios gives us our result. I'm not sure how the law of cosines comes into play, except possibly in proving spherical inversion is conformal.