What does a triangle become?

Question

I'll get to the point.

Imagine a right triangle in 2D.

If you move one of the points (not the 90 degree vertex) very far away, then the corresponding angle will become smaller as the point moves further away.

The 90 degree angle will stay constant. The other angle will become bigger, approaching 90, keeping the total angle at 180, by definition.

What happens of the point is at "infinity," will you be left with two 90 degree angles? This is impossible as those sides would be parallel, and therefore ending up with a seemingly impossible shape (it can't be a triangle anymore).

This similar problem can be applied to different shapes and things, as well.

Thank you very much!

Answer 1

You get a side and two parallel half lines originating from the endpoints. Nothing extraordinary.

In standard geometry, the third vertex does not exist. In projective geometry, it becomes a point at infinity.

Answer 2

In the usual Euclidean plane there are no points "at" infinity, so you can never have two angles of a triangle that are exactly 90 degrees each. All you can do is put the a vertex arbitrarily far away in order to get the second angle arbitrarily close to a right angle.

In a projective plane there are points at infinity, and parallel lines meet at those points, so again no contradiction.

There are other possible kinds of geometry to consider, but I suspect I've already gone a step too far by introducing projective geometry.

Answer 3

Great question. The problem you are having I think is caused because of the fact that "infinity" isn't a place a point can be. We can talk about what happens as a point gets arbitrarily far apart, and as you mentioned, the other angle will get increasingly close to 90 degrees. But what if we decided to add a point at infinity? Well, this might cause some problems because all lines would intersect there(so what does parallel mean?) it ends up though if you add the point at infinity, a plane becomes a sphere(this is called the one-point compactification of the plane) On a sphere it ends up that all triangles have degrees adding up to MORE than 180 degrees, so there wouldn't be anything wrong with your triangle having two right angles.

Hopefully this helps with your thinking about infinity. It is a truly tricky concept to grasp fully.