Demo Notes

Here is a self-guided tour. For each file name below, double-click on that file in the Demos directory, and read here the short description of what is going on.

Scripts:

TrigFuncs.T: A pure demonstration of the geometric meanings of the trigonometric functions. Press the 'Run Script' button, and watch how the colored lines change. The lengths of those lines represent the six common trigonometric functions.

circ.T: A script to convince you that perhaps the area of a circle is pi r-squared. Press the 'Run Script' button, and see how a circle is transformed by rigid motions into something that looks a lot like a rectangle of the appropriate size.

Spirograph.T: Just for fun. Press 'Run Script' and draw a picture that's more complicated than anything you can do with your old plastic spirograph.

Lissijous.T: This shows how a lissijous figure with the x and y coordinates given by sines and cosines of various multiples of a free variable is generated. Press 'Run Script' and watch the result. Then adjust the locations of the R and B points (these correspond to the multiples of sine and cosine), and press the 'Run Script' button again. As the script runs, the red and blue points on the circles run around the circles at uniform (but different) rates. The position of the red point determines the x-coordinate, and the position of the blue point determines the y-coordinate of the final figure.

Proofs:

Ninepoint.T: A proof of the nine-point circle. For any triangle, the feet of the altitudes, the midpoints of the sides, and the points midway between the orthocenter and the vertices all lie on a single circle. Move the vertices A, B, and C of the triangle and note that the nine points stay on a circle (but the circle changes, of course). Then, after you've seen a few thousand examples, step through the proof by pressing the 'Next' button (or the 'Prev' button to go back one step). At each stage of the proof, read the text and check that it's true in the diagram. You can adjust points A, B, and C at any stage in the proof. To start the proof over again, press the 'Start' button.

Squares.T: This isn't a famous theorem, but it's a nice result. Again, press the 'Next' button to step through a proof that the three angles in the diagram add to ninety degrees.

Fagnano.T: A nice proof that in an acute-angled triangle, the orthic triangle is the one of minimum length connecting the sides. Step through the proof (which uses an interesting construction), and then try stepping through the proof again after you have changed the shape of the original triangle to have an obtuse angle (so the theorem should fail in that case). Remember that you can adjust the locations of the vertices at any point in the proof.

Euler.T: This proves that the orthocenter, centroid, and circumcenters of any triangle lie in a straight line. Adjust the triangle to see that it's true, and then step through the proof. Finally, you can use the 'Run Script' button to learn about another interesting relationship among these centers.

Constructions:

Harmonic.T: This is a simple construction of the harmonic series. Given a segment of length 1, construct segments of length 1/2, 1/3, 1/4, 1/5, et cetera.

ExtTangent.T: This leads through a step-by-step construction of the common exterior tangents to two circles. Remember that you can step through the construction using the 'Next' button, and that you can adjust the circle sizes as you go (or before you start) to see how it works. For this construction to work, be sure that the circle on the right is larger than the one on the left.

Heptadecagon.T: A construction of a regular heptadecagon (17-sided figure). Step through with the 'Next' button. Don't worry about why it works, but note that fairly long constructions are easy to do in Geometer.

Miscellaneous:

Peaucellier.T: This is a mechanical linkage that converts circular motion into straight line motion. Adjust the lengths and move the point A. You can also step through a proof with the 'Next' button that shows why the linkage generates a straight line.

Poncelet4.T: This diagram illustrates the quadrilateral version of Poncelet's theorem. If two circles are arranged so that a quadrilateral can be inscribed in one and simultaneously circumscribed about the other, then any quadrilateral inscribed in the larger circle will be circumscribed about the smaller one. Adjust O, the center of the inner circle, and move point A to generate an infinte number of quadrilaterals that satisfy the conditions.

cvvvvv.T: Any five points determine a conic section. In this diagram, move any of the five points and see the unique conic section that does so. A conic section is a circle, ellipse, hyperbola, or parabola.

PythagProofFinal.T: This is a visual proof of the Pythagorean theorem. Just press the 'Run Script' button and it executes.

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