Inversion
Inversion transformation is suitable for situations where there are tangent relationships between multiple circles/straight lines in the question. Using the properties of inversion transformation to solve problems in the inversion space can greatly simplify calculations.
The inversion center point O and the inversion radius R are given. If points P and P' on the plane satisfy:
Point P' is on the ray OP.
|OP||OP'| = R²
Then point P and point P' are said to be inversion points of each other.
Properties
The inversion point of a point outside circle O is within circle O and vice versa; the inversion point of a point on circle O is itself.
The inverse figure of a circle A that does not pass through point O is also a circle that does not pass through point O.
The inversion figure of a circle A passing through point O is a straight line passing through point O. Why is it a straight line? Because a point on circle A is infinitely close to point O, its inversion point is infinitely far from point O.
If two figures are tangent and there is a tangent point other than point O, then their inversion figures are also tangent.