An education in math should begin with the study of triangles. If you're young, and have an option to take Geometry someday, I urge you to take the course. You see, when you enter the real world, geometry is crucial in all types of businesses. For example, street signs, buildings and even marquees, all of these are things in the real world that come in triangular form and to be created and marketed successfully, the geometric aspect needs to be understood. Marquees and gazebos, especially can be geometrically intricate sometimes shaped as triangular prisms and other times more like square or rectangular prisms. Such students might think that I'd know everything about triangles. Au contraire! Mathematicians understand how little they understand. From triangles to infinity, there is much left to know.

4 order-3 triangles, and 23 order-4 triangles. Are any missing?

Another question to consider involves cutting a triangle into *n*
parts with equal area. In this question, topological distinction
is no longer necessary. A scalene triangle can be divided into 2
equal area triangular pieces in 3 ways. It can be divided into 3
equal area triangular pieces in 16 ways. The number of ways to
divide a triangle into 4 equal area triangles is infinite, due to
partition number 16, above. Here are the
16 ways to divide a scalene triangle into 3 equal area triangular pieces.

It's easy to make spreadsheets for integer triangles. You can
use Excel, or the free StarOffice (www.sun.com).
For any rational number *a*, it is easy to find many integer
triangles with an angle of arccos(*a*). For example, here is
a whole list of triangles with sides *a b c* for angle *C*
equivalent to -1/*n* , where *n* = 2 to 21. The angle *C*,
opposite side *c*, is getting increasingly closer to being a
right angle.

Minimal integer triangles of varying degrees

It turns out that integer triangles that have an angle in common are
all closely related. The simplest examples are the Pythagorean
Triangles. Michael Somos has made a table of
Pythagorean Triangles. All are expressed by a formula in p and
q, where a1 = q^{2}-p^{2}, a2 = 2*p*q, a3 = q^{2}+p^{2}.
a1^{2} + a2^{2} = a3^{2}, which gives a right
triangle. Bill Harris has found a formula which
will give infinite triangles with any given angle.

For what rational numbers *a*, *b*, and *c* will
cos(arccos(*a*)+arccos(*b*)+arccos(*c*))=1 ? This
is a surprisingly deep question. Here is an answer by William
Watson. Marek Ctrnact verifies the first is minimal.

What is the smallest triangle, in terms of perimeter, that can be
divided into **4** nonsimilar triangles with integer sides?
Bill Daly was intrigued by the problem. **Some
of the methods Bill Daly used to solve the problem are here**.
Konstantin Knop has found the minimal answer:

Daniel Scher's Geometry in Motion has many interactive problems with triangles. A vaguely connected problem to all this is to divide a 45-degree right triangle into four pieces that can be rearranged into an equilateral triangle. The solution is here.

Noam D. Elkies posted an interesting letter at rec.puzzles:
"There's a shortcut called Stewart's Theorem, for which we learned (at
Stuyvesant Math Team -- those were the days) the mnemonic "a man and his
dad put a bomb in the sink". Misspell "sink" as "cinc",
disemvowel the bomb and the cinc, and get Stewart's Theorem: **man
+ dad = bmb + cnc** . Here a,b,c are sides of a triangle ABC; the "a"
side is divided into m+n by a point at distance d from the opposite
vertex. m and n are labeled so that the products bmb and cnc
involve disjoint line segments. In our case, we have: b=c=7, a=13, m=5,
n=8. So, 5*8*13 + 13 d^2 = 7^2 (8+5) = 49*13 So, d^2 = 49 -
5*8 = 9 and d=3 as desired.

SOHCAHTOACos(A) = Adjacent/Hypotenuse Tan(A) = Opposite/Adjacent |

From Nick Baxter: Find the smallest integer triangle where one angle is double the size of another. Send Answer. Bonus points for finding 3x, 4x, 5x, 6x, and 7x.

Can a triangular prism be fair? Probably not, when tossed either glass or mud. Here's related experimental data that Juha Saukkola pointed me to.