| Class Schedule |
- Similar Triangle Lemma (Review from 444): If A' and B' are
inversions of A and B in a circle with center O, then triangle OAB is
similar to triangle OB'A'.
- Inversion Image of a Line: Suppose that c is a circle with
center O and m is a line.
- Line through O: If m is a line through O,
the inversion image m' of m is the line m itself (except
that the inversion of O is not defined and O is not the inversion
of any point). Proof: Use the Definition of Inversion.
- Line note through O: If m is a line not through O,
then the inversion image m' of m is a circle through O (with
the diameter through O perpendicular to m). Note: m' is the
circle with O removed, since O is not the image of any point on
m. Proof: Use the Similar Triangle Lemma.
- Inversion Image of a Circle: Suppose that c is a circle with
center O and d is a circle.
- If d is a circle through O, then the inversion image
d' of d is a line not through O. Proof: This is the
same as the previous theorem about the image of a line not through
O, using the fact that inversion is its own inverse.
- If d is a circle not through O, then the inversion
image d' of d is a circle not through O. Proof outline: This
proof uses dilation and the power of a point. The key fact is that
the inversion image of the circle is also a dilation image, though
if A is a point in the circle d, the inversion image A' is one point
in d' and in general d*, the dilation image, is another.
- Conformality: The inversion images of two objects meet at the same
angle as the orginal objects.
- Two Lines not through O. If m and n are two lines not through
O that intersect at point P, then the inversion of these lines in
a circle centered at O are two circles m' and n' which intersect
at O and also at P'. The circles meet at the same angle at O and
at P' as the lines meet at P because the tangents to the circles
at O are parallel to m and n.
- Other cases (not proved in class Friday): If m and n are
either lines or circles that intersect at a point P, the inversion
images m' and n' of m and n in a circle c meet at the same angle
at P' as the originals m and n meet at P .
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