意訳をお願いします
8.4.1 Inversion Preserves Angles Between Lines
Suppose AB and CD are two lines in the plane that intersect at the point P. If they
are inverted in a circle with center O, we will show that the resulting gures make the
same angle with each other.
The easiest case is if O, the center of inversion, happens to be the same as P.
Since both lines pass through the center of inversion, they are transformed into
themselves so their images trivially have the same angle between them.
If the center of inversion is not on one of the lines (but it may be on the
other), then the line that does not contain the center will be inverted to a circle
passing throughO. Suppose O is not on AB. Consider the line OX passing
throughO that is parallel to AB . Under inversion, OX is transformed
transformed
into itself (since it passes through the center of inversion), and AB is inverted
into a circle K passing through O. Then OX must be tangent to K, since it touches
K at O, and if it intersected K in more than one place, OX and AB would intersect,
which is impossible since they are parallel.
If O is on CD, then CD is inverted to itself, and its angle with OX is clearly
unchanged (OX is also inverted into itself). But CD makes the same angle with OX
as with AB since they are parallel, so the inverses of AB and CD make the same
angle.
Finally, if O is on neither line, then both AB and CD are inverted to circles that
meet at O, and the lines tangent to those circles at O (OX and OY in the gure) are
parallel to the original lines, so they make the same angles as did the original lines.
お礼
有難うございます、ほっとしました。