# The Final Pathetic Bleatings of the Forum

Question:

This is a question mainly for the mathematicians and
scientists(although anyone else should feel free to answer.)
I have to do a term paper on some mathematical subject. I
have to be able to explain it to a pre-calculus class, so it
can't be too complex for the nimrods. What topic do you
suggest I choose?

Replies:

Like many knowledge workers, I yearn to be a thing worker.

I suggest that you study triangles. There are millions and millions of interesting facts about triangles.

For example, you could study the nine-point circle. This is the circle that passes through mid-points of the sides, the legs of the altitudes and the mid-points of line segments beween vertices and the orthocenter.

Ok, so, did you know that the mid-point of a side is also the mid-point of the line segment determined by the points of tangency of the side with the in-circle and the ex-circle. A picture would make this clear.

Do it on tesseracts and their relationship to time travel.

If you trisect the angles of the triangle then the intersections of the trisectors form an equilateral triangle.

If you form equilateral triangles above the sides of a triangle, their centers form an equilateral triangle, known as Napoleon's triangle.

Even more, if you flip the triangles, their centers still form an equilateral triangle.

In fact, there are 27 Napoleon triangles.

The in-circle of a triangle and the circum-circle of the mid-points of the sides touch.

A triangle is isosceles if, and only if, the center of the incircle lies on the Euler's line.

Definition: Choose a point inside a triangle. Drop perpediculars to the sides. The resulting triangle is called the pedal triangle of the point.

Theorem: The pedal triangle of the pedal triangle is similar to to the original triangle.

Arrr!

A side bisects the segment between the orthocenter and the intersection of the altitude on that side with the circumcircle.

Take all chords passing through a given point inside a circle (different from the center). For each one of them, construct the intersection of the tangents at the vertices of the chord. The constructed points are colinear. This is cool.

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