Definition: Punctured Plane

Let E be the extended Euclidean plane. Let be a particular point in E. We will call ={in E | is not } the punctured plane.

Definition: new-line

Let C be any extended circle in E that passes through . A new-line, n, is a set of the form:

A new-line segment is an arc of C which does not contain . (If C is of the form l union {} where l is an ordinary line, then a segment is an ordinary segment of l not containing , or the union of the point {} with two oppositely directed rays emanating from the end points of the new segment, neither of which contain ).

Can you guess how a new-ray should be defined?

Theorem: Two Points Determine a New-Line

Given two distinct points of , there is exactly one new-line that passes through them.
Proof:

Corollary

Two distinct new-lines intersect in at most one point.

Definition: Parallel

Two new-lines are said to be parallel if they do not intersect.

Theorem: Unique Parallel

Given a new-line, n, and a point, P, not in n, there is exactly one line through P parallel to n. (Problem 6)

Corollary

If a,b,c are distinct new lines and a||b and b||c, then a||c.

Proof

Use contradiction. If a,b,c are new lines and a||b and b||c, but a is not parallel to c, then a and c intersect at some point, P. P is not in b since P is in a and c and neither a nor c intersect b. But we now have two distinct new-lines, a and c, passing through P both of which are parallel to b. This contradicts the theorem just proved.

Definition: Perpendicular

Two new-lines are called perpendicular if they meet at right angles.

Theorem: Unique Perpendicular

Given any point, P, in and any new-line, n, there is a unique new-line, m, passing through P that is perperndicular to n. (problems 1 and 2)
Proof:

Definition: New Circle

A new-circle, C, with center S in passing through P in is defined by:

C={P' in| P' is the inverse image of P with respect to some new-line, n, that passes through S.}

Theorem: New Circle

A new-circle is an ordinary Euclidean circle and is orthogonal to every new-line passing through its center, and the center of the new-circle in not the same as the ordinary Euclidean center of the ciricle. (problem 5).

Theorem: Orthogonal Transversals

Let m, n be two parallel new-lines. Let and be any two points on m. Drop perpendiculars from both of these points to n. Let and be the two points on n where the perpendiculars from and meet n. Then and are inverses of each other with respect to some new line, i. (problem 8)
Proof:

Note:

This theorem says that any "measuring rod" separating two parallel new-lines at one spot can be transformed into the "measuring rod" separating the lines at any other spot by a "reflection".

Definition: New-Triangle

Given any three points, A, B, and C, in , a new-triangle consists of three new-line segments: AB, BC and CA.

Theorem: Angle Sum for New-Triangle

The sum of the angles of a new- triangle is a straight angle (180 degrees). (problem 3)