Tag Archives: chord

GEOSTRUCT, a program for investigative geometry

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.

help pic 1
Two lines, with points placed on them
help pic 3
Three random lines with two points of intersection generated
help pic 6
Five free points, three generated circles and a center point
help pic 7
Three free points, connected as point pairs, medians generated
help pic 5
Two free circles and three free points, point pairs and centers generated
gif line and circle
GIF showing points of intersection of a line with a circle
hypocycloid locus
Construction for locus of hypocycloid
circle in a segment
GIF showing a dilation (stretch) in the horizontal direction
gif piston cylinder
Piston and flywheel
gif touching2circles
Construction for circle touching two circles
gif parabola
Construction for the locus of a parabola, focus-directrix definition.







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Calculus without limits 3

So you tried, or you didn’t, now here is the derivative of 1/sqr(x)


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Calculus without limits 2

As h approaches zero
I quietly despair.
It really is the limit.
Please don’t take me there.

The funny thing about the calculus is that it was brought into existence by Isaac Newton in 1666 or earlier, and was developed and used without the idea of limits for over 150 years. The first attempt to get rid of the troublesome infinitesimals was by Cauchy in 1821, where he introduced the chord slope (f(x + h) – f(x))/h. The whole business of finding a satisfactory definition of the derivative was finally achieved by Weierstrass in the mid 19th century.

So here we go with cubics, and the same approach can be used for any whole number power of x, even negative ones. You should try it.


Next time  sin(x) and cos(x), so no more  sin(h)/h stuff.

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Calculus without tears (that is, without limits)

“As h approached zero I reached the limit of my understanding.”

So it seemed to me that calculus without limits would be a good idea.

Not just for powers of x, but also for trig, exp and log functions.

This is the first of several posts on this subject.

calculus1 pic


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