Consider a circle (c) and a point P. To construct the [polar] of P with respect to (c) (see Polar.html for the definition), take two arbitrary points A, B on c and draw the cords through these points: [APC] and [BPD]. Then locate the intersection points E, F of the opposite sides of the resulting cyclic quadrilateral ABCD. Line [EF] is the polar of P.
The proof is an immediate consequence of the basic figure for the construction of harmonic conjugate points discussed in Harmonic.html . By the discussion there we know that points P, G are harmonically conjugate to B, D and points P, H harmonically conjugate to A, C hence the definition of the polar applies for line [EF]. This procedure of constructing the polar works also for conics. The preceding proof transfers verbatim to that case.
For a further property of the polar, concerning the duality of the correspondence pole <--> polar see the file Polar3.html .
![[alogo]](alogo.png){kind=small}
Construction of the polar ![[0_0]](Polar2_files/Polar2_0_0_0.png)
![[1_0]](Polar2_files/Polar2_0_1_0.png)