Math 302

Journal on Weird Triangles.

Here is one possible definitions of triangle : Take any three points a, b, and c. Now choose any line segment connecting a and b, any connecting b and c, and any connecting c and a.

Can you think of any "triangles" on the Euclidian plane that satisfy this definition but seem weird? Will there be any strange triangles on the hyperbolic plane? Next, try to find as many strange triangles on the sphere as you can. There are quite a few. If you have time, try this for the cylinder too.

The best way to do this is to first pick any three points. Try some strange things, like making all three points lie in a line, or choosing two to be antipodal. Then, pick any three segments connecting them. Make sure that you don't always pick the shortest segment! Do you want to change your definition? What do you think would be a good definition for triangle?

Now, what does it mean for two triangles to be similar ? There are alot of possibilities, here are a few things to look at.

• The set of points on the interior of the triangle.
• The set of points you make when you draw the triangle.
• The vertices of the triangle.
• The vertices plus the segments of the triangle.