Exercise 37.12
- Author:
- Gianluca Occhetta
Verify the following 5-step construction for the inverse of a point A with respect to the circle Г. Take a circle of any radius with center A, to meet Г at Ρ and Q. Let AP and AQ meet Γ in further points R, S. Join PS and RQ.
Their intersection is A'. (This works equally well if A is inside Γ.)
We first contruct A' by considering the circle passing by OPR - which is the image of the line AR via the inversion - and intesecting it with the lie AO.
Then we must show that R, A' and Q are collinear.
The angles ORP and OAR are equal, since they stan on equal chords.
The angle PA'A is equal to ORP since A'ORP is a cyclic quadrilateral.
The angles APA' and AQA' are equal by simmetry, so the angles OA'R and QA'A are equal and R, A' and Q are collinear.