Essay Ideas

Last modified on January 7, 2014

Valentine Hearts

Friday the 13th.

Two problems. First, show there are at most three and at least one Friday the 13th in each year. And, second, consider what years between 1990 and 2110 can have two consecutive months can be have Friday the 13th.

Gingerbread Man.

This is a Microsoft Excel file to illustrate iteration 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.   To be explored:

     Is Napoleon's triangle equilateral?

     There are two Napoleon triangles, one with the equilateral triangles constructed outward and one with the equilateral triangles constructed inward.

     The difference in the areas of the two Napoleon triangles is the same as the area of the original triangle.

     Show that the three circumcircles of the equilateral triangles constructed on the sides are CONCURRENT at a single point.     Show that this concurrency point is also the point of concurrency of the lines from each vertex of the original triangle to the external vertex of the equilateral triangle constructed on the opposite side.

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.


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.


The inversion of a point P in a circle of radius r, center at O, is a mapping of  P to a point P' such that OP.OP' = .   If   r = 1, we describe this mapping as inversion in the unit circle.

Click HERE for a GSP file that has a Script Tool for inversion of a point P in a circle.   The tool works for locating the point P either outside or inside the circle of inversion.








Some Explorations with Sequences.

Rotate Triangle.

What is the locus 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. . .

Next, explore a similar animation where  P  is animated about a circle.

Triangle Constructions


The tangrams "puzzle" has been used in mathematics classes from middle school on up to pose problems with "hands on" work for geometric relationships.   The problems use the seven pieces of the tangram and arrange them to form other shapes or images.   There are 13 convex shapes but a very large number of non convex shapes.    A google search will quickly show the richness of the tangram puzzle.


Three Circles.

Two problems are proposed involving three circles tangent 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.

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