One kind of zero-dimensional object

• E-points

Two kinds of 1-dimensional objects

• E-lines
• E-circles

One kind of zero-dimensional object

• I-points: any E-point or also Inf = Infinity

One kind of 1-dimensional object

• I-circle: any E-circle or any E-line, with the understanding that Inf is considered a point on the line.

Two kinds of "reflection"

• Line reflection in a line m.
• Inversion in a circle m.

One kind of "reflection"

• Inversion in an I-circle m (which is E-reflection if m is an E-line and is E-inversion if m is an E-circle)

Theorems about "reflection images"

• If m is an E-line and d is an E-line, then the reflection of d in m is an E-line.
• If m is an E-line and d is an E-circle, then the reflection of d in m is an E-circle.
• If m is an E-circle and d is an E-line through the center of m, then the reflection of d in m is an E-line.
• If m is an E-circle and d is an E-line not through the center of m, then the reflection of d in m is an E- circle.
• If m is an E-circle and d is an E-circle through the center of m, then the reflection of d in m is an E-line.
• If m is an E-circle and d is an E-circle not through the center of m, then the reflection of d in m is an E- circle.

Theorems about "reflection images"

• If m is an I-circle and d is an I-circle, then the inversion of d in m is an I-circle.

Angle invariance and orthogonality

• If m is an E-circle or E-line and d and e are E-circles and/or lines, then the angle(s) between d and e are the same as the angle(s) between the images of d and e by inversion or reflection in m.
• In particular, two orthogonal I-circles reflect or invert to two orthogonal I-circles.
• If m is an E-line and d is an E-line, then the reflection of d in m is the same as d itself, if either d = m or d is orthogonal (perpendicular) to m.
• If m is an E-line and d is an E-circle, then the inversion of d in m is the same as d itself, if d is orthogonal to m (i.e., m is a diameter of d).
• If m is an E-circle and d is an E-circle, then the inversion of d in m is the same as d itself, if either d = m or d is orthogonal to m.
• If m is an E-circle and d is an E-line, then the inversion of d in m is the same as d itself, if d is orthogonal to m (i.e, d is a diameter of m).

Angle invariance and orthogonality

• If m is an I-circle and d and e are I-circles, then the angle(s) between d and e are the same as the angle(s) between the images of d and e by inversion in m.
• In particular, two orthogonal I-circles invert to two orthogonal I-circles.
• If m is an I-circle and d is an I-circle, then the inversion of d in m is the same as d itself, if either d = m or d is orthogonal to m.