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.

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
iL(M) = M if and only if L and M are orthogonal.

Proof of the Mirror Theorem.

Main inversive page