Let a point A be on the polar of a point P with respect to the circle (c) with center O. Consider also the polar (a) of A with respect to (c) and the intersection point Q of (c) with line OP. Then, for every point B on (a) construct triangle QCD and extend QC, QD to intersect (p) at I, j. Quadrangle CDJI as shown is cyclic.
Consider the inversion which interchanges line (p) with circe (c). Its circle (q) is centered at Q and its radius satisfies r2 = QL*QK. By this inversion points C, D are transformed to I, J respectively and by the properties of inversions CDJI is cyclic and its cicumcircle is invariant under the inversion (is orthogonal to (q) and maps to itself). Some other relations here are worth noticing: [1] The inversion maps the circumcircle (d) of triangle QIJ to line CD. Thus (q) and (d) intersect on line ACD. [2] The intersection point E of (c) and (d) maps by the inversion to A (intersection of CD and (p)).
![[alogo]](alogo.png){kind=small}
Polar and inversion ![[0_0]](PolarAndInversion_files/PolarAndInversion_0_0_0.png)
![[0_1]](PolarAndInversion_files/PolarAndInversion_0_0_1.png)
![[0_2]](PolarAndInversion_files/PolarAndInversion_0_0_2.png)
![[1_0]](PolarAndInversion_files/PolarAndInversion_0_1_0.png)
![[1_1]](PolarAndInversion_files/PolarAndInversion_0_1_1.png)
![[1_2]](PolarAndInversion_files/PolarAndInversion_0_1_2.png)
![[2_0]](PolarAndInversion_files/PolarAndInversion_0_2_0.png)
![[2_1]](PolarAndInversion_files/PolarAndInversion_0_2_1.png)
![[2_2]](PolarAndInversion_files/PolarAndInversion_0_2_2.png)