compass and straightedge construction of inverse point
Let $c$ be a circle in the Euclidean plane^{} with center $O$ and let $P\ne O$. One can construct the inverse point^{} ${P}^{\prime}$ of $P$ using compass and straightedge.
If $P\in c$, then $P={P}^{\prime}$. Thus, it will be assumed that $P\notin c$.
The construction of ${P}^{\prime}$ depends on whether $P$ is in the interior of $c$ or not. The case that $P$ is in the interior of $c$ will be dealt with first.

1.
Draw the ray $\overrightarrow{OP}$.

2.
Determine $Q\in \overrightarrow{OP}$ such that $Q\ne O$ and $\overline{OP}\cong \overline{PQ}$.

3.
Construct the perpendicular bisector^{} of $\overline{OQ}$ in order to find one point $T$ where it intersects $c$.

4.
Draw the ray $\overrightarrow{OT}$.

5.
Determine $U\in \overrightarrow{OP}$ such that $U\ne O$ and $\overline{OT}\cong \overline{TU}$.

6.
Construct the perpendicular bisector of $\overline{OU}$ in order to find the point where it intersects $\overrightarrow{OP}$. This is ${P}^{\prime}$.
Now the case in which $P$ is not in the interior of $c$ will be dealt with.

1.
Connect $O$ and $P$ with a line segment^{}.

2.
Construct the perpendicular bisector of $\overline{OP}$ in order to determine the midpoint^{} $M$ of $\overline{OP}$.

3.
Draw an arc of the circle with center $M$ and radius $\overline{OM}$ in order to find one point $T$ where it intersects $C$. By Thales’ theorem, the angle $\mathrm{\angle}OTP$ is a right angle^{}; however, it does not need to be drawn.

4.
Drop the perpendicular^{} from $T$ to $\overline{OP}$. The point of intersection is ${P}^{\prime}$.
A justification for these constructions is supplied in the entry inversion of plane.
If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.
Title  compass and straightedge construction of inverse point 

Canonical name  CompassAndStraightedgeConstructionOfInversePoint 
Date of creation  20130322 17:13:17 
Last modified on  20130322 17:13:17 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  12 
Author  Wkbj79 (1863) 
Entry type  Algorithm 
Classification  msc 51K99 
Classification  msc 53A30 
Classification  msc 51M15 