Consider an arbitrary logarithmic spiral of growth rate $q$ per angle $\theta$ and flair coefficient $b=\ln q/\theta$. Plot the spiral $z=e^{(b+i)\theta}$ and mark off the points that are equally spaced in $\theta$. Connect the points sequentially and build out isosceles triangles of vertex angle $\theta$ from each line segment. See the figure below for some examples.
I find that all the triangles and their circumscribed circles are collinear upon inversion, say $w=1/z^*$ where $z$ is either a triangle or circle.
I understand that circles can be mapped to straight lines by Möbius transform, but I don't understand how that translates to this problem. This one transform that I found, namely,
$$w=\frac{iz+i}{-z+1}$$
did not seem to work. Moreover, it is not equal to $1/z^*$. But more to the point, what makes the logarithmic spiral gnomons have this property?
Saying that your circles are transformed into straight lines by an inversion with center (pole) $O$ is equivalent to say that your circles are passing through the origin.
Therefore, the issue is reduced to establish that the origin belongs to the circumscribed circles to the isoceles triangles ; this is a rather simple consequence of the inscribed angle theorem as we can see on the figure below better than with words.
Besides, on your figures, especially the second one, we see a discrete approximation of the evolute of the spiral (itself a spiral) defined either as the envelope of its normals or as the locus of centers of curvature. This could constitute another approach to your issue.