1. The inversion center has no inversion point. The two pairs of inversion points of a straight line are * * * circles, which are orthogonal to the inversion base circle. The inverse image of a circle orthogonal to the base circle is the original circle.

2. Change φ through inverse evolution, and turn any straight line passing through the inversion center O into itself. In other words, any straight line passing through the inversion center is an invariant figure replaced by this inversion. (straight line → straight line)

3. The inverse evolution of φ changes any straight line that does not pass through the inversion center O into a circle that passes through the inversion center O, and the tangent of this circle at point O is parallel to the straight line. (straight line → circle)

4. Change any circle passing through the inversion center O into a straight line that does not pass through the inversion center O, and this straight line is parallel to the tangent of the circle passing through the point O (circle → straight line).

Note: Property 3 and Property 4 are contradictory propositions.

5. The circumference that does not pass through the inversion center O is changed into the circumference that does not pass through the inversion center O by inverse evolution. (circle → circle)

Since a straight line can be regarded as a circle, the above properties 2-5 can be integrated into anti-evolution, and a (generalized) circle can be changed into a (generalized) circle. This theorem is usually called the cyclicity of inverse transformation.

6. The included angle of any two straight lines at the intersection point A is equal to the included angle of their reflection maps at the corresponding point A', but the directions are opposite.

7. The included angle of two intersecting circles at the intersection point A is equal to the included angle of their inverted graphs at the corresponding point A', but the directions are opposite.

8. The angle between a straight line and a circle at the intersection point A is equal to the angle between their inverted images at the corresponding point A', but the directions are opposite.

The above properties 6-8 can be summarized as the angle between two intersecting (generalized) circumferences at the intersection point A, which is equal to the angle between their inverse (generalized) circumferences at the corresponding point A', but in opposite directions. Theorem 2 is called the inverse conformal property of inverse evolutionary transformation.

Because the inverse evolutionary transformation has the properties of preserving circle and inverse conformal mapping, it has become an important tool for proving problems and drawing. According to Theorem 1 and Theorem 2:

9. The inverse images of two orthogonal circles are still orthogonal.

10, the inverse images of two tangent circles are still tangent. If the tangent point happens to be the inversion center, the inversion image is two parallel lines.

Negative power transformation can be transformed into the product of positive power transformation and reflection about anti-pole.