I am reading(adsbygoogle = window.adsbygoogle || []).push({}); Visual Complex Analysisby Dr. Tristan Needham and am hung up on some of the geometrical concepts. In particular, I am having trouble with ideas involving the geometric properties of numbers like:

[itex]\frac{z-a}{z-b}[/itex]

Note: I am still in the first and second chapters, which deal with the basic geometry of complex numbers and functions.

As an example problem, here is one I've been trying to figure out for a few days:

As an attempted solution, I chose three arbitrary pointsa, b, zand constructed the perpendicular bisector ofaandband a three point circle througha, bandz. The chord throughaandbdivides the circle into two regions.

Sincezis a variable under a constraint, it may move freely about this circle. As long aszstays on the same side of chord [itex]\stackrel{\rightarrow}{ab}[/itex], the angle between the chords [itex]\stackrel{\rightarrow}{az}[/itex] and [itex]\stackrel{\rightarrow}{bz}[/itex] will be a constant. I interpreted (but have found no way to prove) the constant angle as the constant referred to in the problem.

That being said, it looked to me like the tangent of this angle was equal to [itex]Im((z-a)/(z-b))/Re((z-a)/(z-b))[/itex],

whereIm(z)andRe(z)are the imaginary and real parts of the complex numberz.

This is as far as I got. I showed that the angle between chords was constant along the three point circular arc ofa,bandz. I have no idea if this is even the right place to look, and this problem is only one of many dealing with complex ratios that I simply don't understand.

Any help?

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# A Locus in the Complex Plane

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