If the author of the topic is interested in a scientific definition:
As a mathematician, I'll say:
A triangle is a geometric figure consisting of three points that do not lie on the same line, and three line segments that connect these points in pairs.

Regarding touching it, of course, you can! Why not? For example, a ruler in the shape of a triangle. It's still a triangle!

I disagree here – you can touch a ruler in the shape of a triangle, anything that resembles a triangle in shape. But there's one caveat:
if you can touch it, then it has height, length, and width (you can even calculate time), which a figure in a plane cannot have!
And a ruler is called a triangle for simplicity.
(this is my opinion, which is what the dispute is about...)

strange question) if you mean a triangular ruler, called a triangle, then, without a doubt, you can) if you imagine a triangle in your mind, then you can only touch it mentally. I don't know what else to say.

No, let's call things by their names: a ruler is a ruler, a triangle is a figure in a plane, otherwise you could call a pyramid a triangle :D
Regarding touching it in your mind – I specifically chose the word "touch" to emphasize materiality