I'm reading Silverman's Complex variable with application.
at page 78, the author says
"We say that $s$ and $s^*$ are the inverse points with respect to circle in $\mathbb{C}$ if every line or circle passing through $s$ and $s^*$ intersect at right angles."
and the author consider the case of unit circle and a point $z$ lying inside the unit citcle. And he claimed that the inverse point of $z$ is $\frac{1}{\bar{z}}$. He show this as follows:
Let $L$ be a line passing through the center $O$ (the origin) and $z$. Draw a line $S$ perpendicular to $L$ through $z$. This line intersects unit circle twice. Draw tangent lines at which $S$ intersects the circle. Then the two tangent line intersect at the line $L$. The point on the $L$ at which two tangent line intersect is $\frac{1}{\bar{z}}$. So $O$ and $z$ and $\frac{1}{\bar{z}}$ are collinear.
My questions are like this:
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What did he mean by "every line or circle passing through two inverse points intersect at right angles"? What are the objects which intersect each other? If you want to say "intersect", there has to be at least two objects. Does it mean that given circle and any arbitrary circle or line passing through two inverse poinrs intersect orthogonally? I can't imagine how it might look like.
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and The phrase "Every line passing through two inverse points" looks weird at least to me, since if we have two points on the plane, the line passing through those points is uniquely determined. Why did he talk like that?
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Why if you follow the above construction then come out $\frac{1}{\bar{z}}$? I don't get it. Actually I'd like to see the algebraical construction corresponding above geometrical construction. (e.g. starting from setting z and arrive at $\frac{1}{\bar{z}}$)
Best Answer
It might be easier to start with the converse statement. If a circle $C'$ through $z$ and $z^*$ intersects the unit circle $C$ at $z_1$, then $|z_1|^2 = 1 = |z| \, |z^*|$, therefore the radius of $C$ at $z_1$ is a tangent to $C'$ by the tangent-secant theorem. The construction with the line $S$ follows because if $S$ intersects $C$ at $z_1$, the segment $[z_1, z^*]$ is a diameter for the circle through $z, z^*, z_1$.
"intersect at right angles" is supposed to be "intersects the unit circle at right angles" and "tangents at which $S$ intersects the unit circle" is supposed to be "tangents at the points at which $S$ intersects the unit circle". And $|z - z_0| \, |z - z^*| = R^2$ should be $|z - z_0| \, |z^* - z_0| = R^2$.