Complex Bashing

Hi ! I have a doubt regarding complex numbers in geometry ( a.k.a. bashing).

In this wiki, it is written that points A,B,C are collinear iff,

$\frac{a-b}{\ \overline{a}-\overline{b}\ }=\frac{a-c}{\ \overline{a}-\overline{c}\ }.$ or equivalently, $\frac{a-b}{b-c}$ is real.

But I am struggling with the reasoning / proof. So it'll be really nice if someone can help me out with the proof.

Note : I currently know :-

representation complex numbers as Cartesian and polar coordinates
rotation of complex numbers by a given angle
translation of a number to another point

#Geometry

Easy Math Editor

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Comments

$\large{\frac{a - b}{\bar{a} - \bar{b}} = \frac{a - b}{\bar{(a-b)}} = \frac{M \angle \theta}{M \angle - \theta} = 1 \angle 2 \theta}$

In the above, $\theta$ is the angle of the line segment from B to A with respect to the horizontal. If we do the same thing with the C and A vector coordinates, and it yields the same angle value, the three points must be collinear.

Steven Chase - 2 years, 6 months ago

If A, B, and C are colinear then we can simply take one point and scale it by a real number to get the other points. For example, we take A and scale it to get the other two points: $B = xA$ and $C = yA$ , where $x$ and $y$ are real. Then we have

$\frac{a-b}{b-c} = \frac{a-ax}{ax-ay} = a \frac{1-x}{x-y}$

Since $x$ and $y$ are real numbers, then the quotient above is real.

Levi Walker - 2 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$