# Monthly Archives: August 2017

## Parabola, it’s scarily simple…

No distances, no circles, and you can easily derive an equation.

Just a right angled triangle.

First, the definition of a parabola from the focus and directrix.

Pick a line, the directrix, and a point (B) not on that line (the focus): Find the line at right angles, passing through a point (C) on that line. Now find the line from B to C, and the midpoint of BC, which will be D. Find the line at right angles to BC from D, and the intersection of this line and the vertical line, E, is a point on the parabola. As point C is moved the parabola is traced out. The picture is completed with the line BE. Check it!

## Inversion in a circle

Diagram, then text: The circle has radius 1 and centre at the origin.
The line is x = a

Now 1/a is the inverse of a, so a * 1/a = 1, and is fixed.
The line z from (1/a,0) to (x,y) is orthogonal to the radial line, so r/z = Y/a and the two triangles are similar
and r/(1/a) = a/R
Hence rR = a * (1/a) = 1, and both conclusions are true.

The point (x,y) follows a circular path, and rR = 1

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