Equal incircles theorem

The theorem states that the incircles of the triangles formed (starting from any given ray) by every other ray, every third ray, etc. and the base line are also equal. The case of every other ray is illustrated above by the green circles, which are all equal.


From the fact that the theorem does not depend on the angle of the initial ray, it can be seen that the theorem properly belongs to analysis, rather than geometry, and must relate to a continuous scaling function which defines the spacing of the rays. In fact, this function is the hyperbolic sine.


The theorem is a direct corollary of the following lemma:


Suppose that the nth ray makes an angle ${\displaystyle \gamma _{n}}$ with the normal to the baseline. If ${\displaystyle \gamma _{n}}$ is parameterized according to the equation, ${\displaystyle \tan \gamma _{n}=\sinh \theta _{n}}$, then values of ${\displaystyle \theta _{n}=a+nb}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are real constants, define a sequence of rays that satisfy the condition of equal incircles, and furthermore any sequence of rays satisfying the condition can be produced by suitable choice of the constants ${\displaystyle a}$ and ${\displaystyle b}$.