Wilson Stothers' Inversive Geometry and CabriJava Pages

Orthogonal i-lines

The Uniqueness Theorem
If A and B are distinct points on an i-line L, then there is a unique i-line M
through A and B orthogonal to C.

Since the proof is a good example of
the use of inversion, we give it in full.

Proof
Since A and B are distinct, one of them (A say) is not Ñ.
Let C be a circle with centre A, so L passes through the centre of C.
Let M be any i-line through A and B.
Let B', L', M' be the inverses of B, L, N with respect to C.
By the Invariance Theorem, L and M are orthogonal if and only if
L' and M' are.
By the Inversion Theorem, L' and M' are extended lines.
As B is on L and M, the inverses pass through B'.
But there is exactly one i-line, N say, through B' perpendicular to L'.
Hence there is a unique M (the inverse of N with respect to C), as required.
Note the structure of the proof.

We transform the picture to get a single, easy case.
We verify that the result in the special case implies the result in the original picture.

In a similar way we can prove a result fundamental to the creation of hyperbolic geometry.
The Mirror Theorem
Suppose that L and M are i-lines, then
i_L(M) = M if and only if L and M are orthogonal.
Proof of the Mirror Theorem.

Main inversive page

The Uniqueness Theorem If A and B are distinct points on an i-line L, then there is a unique i-line M through A and B orthogonal to C.	Since the proof is a good example of the use of inversion, we give it in full.
Proof Since A and B are distinct, one of them (A say) is not Ñ. Let C be a circle with centre A, so L passes through the centre of C. Let M be any i-line through A and B. Let B', L', M' be the inverses of B, L, N with respect to C. By the Invariance Theorem, L and M are orthogonal if and only if L' and M' are. By the Inversion Theorem, L' and M' are extended lines. As B is on L and M, the inverses pass through B'. But there is exactly one i-line, N say, through B' perpendicular to L'. Hence there is a unique M (the inverse of N with respect to C), as required. Note the structure of the proof. We transform the picture to get a single, easy case. We verify that the result in the special case implies the result in the original picture.
In a similar way we can prove a result fundamental to the creation of hyperbolic geometry. The Mirror Theorem Suppose that L and M are i-lines, then i_L(M) = M if and only if L and M are orthogonal. Proof of the Mirror Theorem.