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!
The row operation matrix inversion method is so neat and ingenious, and it has the same operations for all dimensions of matrix.
Here is a step by step approach, where firstly the dimension is chosen, then the first of the buttons is selected (Start). After which the buttons are selected in order. The states of the left and right matrices are displayed at each stage, and finally the identity matrix appears on the left, and the inverse of the original matrix appears on the right.
The first display shows the original matrix on the left. Nothing has been done yet.
The second one is the 2 x 2 matrix inverted.
The third is the 4 x 4 matrix inverted.
This is the standard way of inverting a matrix. I came across this first in 1960. Oh so long ago! Goodbye Cramer’s Rule.
I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.
It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.
Just try it out, it’s free.
Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html
.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.
Two lines, with points placed on them
Three random lines with two points of intersection generated
Five free points, three generated circles and a center point
Three free points, connected as point pairs, medians generated
Two free circles and three free points, point pairs and centers generated
GIF showing points of intersection of a line with a circle
Construction for locus of hypocycloid
GIF showing a dilation (stretch) in the horizontal direction
Piston and flywheel
Construction for circle touching two circles
Construction for the locus of a parabola, focus-directrix definition.