Articles of inversive geometry

Constructing the circle inversion inverse of a point with ruler only

I’ve been reading a bit about inversive geometry, particularly circle inversion. The following is a problem from Hartshorne’s classical geometry, which I’ve been struggling with on and off for a few days. I figured it would be helpful to show that $TU\perp OA$ first. At best, I tried to label angles according to which ones […]

Form of most general transformation of the upper half plane to the unit disk.

In David Blair’s book on Inversion Theory, he write that the transformation $$ T(z)=e^{i\theta}\frac{z-z_0}{z-\bar{z}_0} $$ is the most general transformation mapping the upper half plane to the unit circle, provided $z_0$ is in the upper half plane. If $z_0$ is in the lower half plane, then the upper half plane is mapped to the exterior […]

Interesting circles hidden in Poncelet's porism configuration

This question is an investigation starting here, with a straightedge and compass construction of $ABC$ given $(R,r,h_A)$. The key lemma is the following one: Let $\Gamma$ be a circle with centre $O$, radius $R$ and let $A\in\Gamma$. Let $H_A$ be any point inside $\Gamma$ such that $AH_A=h_A$ and let $B’,C’$ be the intersections between $\Gamma$ […]

Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

While browsing around about problems similar to the problem of Apollonius, I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle tangent to three given circles, but one can construct a circle through any three points, tangent to any three lines, passing through […]

Task “Inversion” (geometry with many circles)

Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. –°ircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ in point $K$. Point $M$ is midpoint of $BC$, $L$ midpoint of $B_{1}C_{1}$. –°ircumcircle of triangle $KA_{1}M$ intersects second time $\omega$ in point $T$. Prove, that […]

A circle with infinite radius is a line

I am curious about the following diagram: The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of ‘circle’ and ‘line’ does this hold? Thanks!

Finding the circles passing through two points and touching a circle

Given two points and a circle, construct a/the circle through the two points and touching the given circle. I came across this problem in History of Numerical Analysis by H. Goldstein. I spent some time on this. I have a method of constructing it using radical axis. I am wondering if there is a more […]