EMAT 4690/6690

Essay Ideas

Last modified on February 25, 2008

Valentine Hearts

Two problems. First, show there are at most three and at least one Friday the 13th in each year. And, second, consider when two consecutive months can be have Friday the 13th.

Gingerbread Man.

This is a Microsoft Excel file to illustrate interation of a seed point (x(0), y(0)) by the sequences x(n+1) = 1 - y(n) + abs(x(n)) and y(n+1) = x(n). This point is iterated 500 times and the points plotted in an XY scatter plot. Explore the iteration of various points (x,y) by seeding the initial value of x in cell A1 and the initial value of y in cell B1.

Napoleon's Triangle.

Given any triangle ABC, construct equilateral triangles on each side and find the center of each equilateral triangle. The triangle formed by these three centers is Napoleon's Triangle. This is a GSP sketch.

A circle tangent to a given line and a given circle.

Construct a circle tangent to a given line and a given circle. What is the locus of the center of all such circles? Proof? A GSP Sketch.

Bullseye. An Interesting GSP Sketch by Pam Turner. What did she do? Double click "Animate" to run the animation. Notice the end patterns after the animation runs a minute or two. The envelopes of circles define cardioids.

Inversion. The inversion of a point P in a circle of radius AB, center at A, is a mapping of C to a point C' such that AC.AC' = AB.AB.

Some Explorations with Sequences.

Rotate Triangle. What is the locis of the third vertex of a triangle when its first two vertices are moved along the x and y axes respectively?

Pappas Areas. Take any triangle and construct arbitrary parallelograms on each side. Extend the two outer sides of the constructed parallelograms to where they meet. Use the segment of length d from this point to the near vertex of the triangle to define a length and direction for constructing a parallelogram on the third side. Show that the sum of the areas of the first two parallelograms is equal to the area of the third.

Polygonal Path. This is a Geometer's Sketpad animation where a point P is animated about a polygonal path. An envelope of circles is traced where P is the center of each circle and each circle passes through a fixed point S. Explore. . .

Triangle Constructions

Tangrams. The tangrams "puzzle" has been used in mathematics classes from middle school on up to pose problems with "hands on" work for geometric relationships.

Three Circle. Two problems are proposed involving three circles tanget to a common external tangent and having additional conditions on their constructions.

Trisections of the areas of triangles. Three problems are proposed pertaining to dividing the area in a triangle into three equal parts.

Bisecting the area of a triangle.

The Gergonne Point of a Triangle. Given a triangle ABC with an incircle that is tangent to BC at D, to AC at E and to AB at F. Prove that AD, BE, and CF are concurrent. The point of concurrency is the Gergonne Point.

Send e-mail to jwilson@coe.uga.edu

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