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):

parabola 1

Find the line at right angles, passing through a point (C) on that line.

parabola 3

Now find the line from B to C, and the midpoint of BC, which will be D.

parabola 2

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.

parabola 4

As point C is moved the parabola is traced out.

parabola 5

The picture is completed with the line BE. Check it!





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Inversion in a circle

Diagram, then text:

inversion line and circle

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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This is a reblog.

Do read it.

It’s quite short!


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