Iv'e been given a problem as part of my undergrad degree and I'm really struggling. Here's the question:

Consider for an arbitrary triangle the possibility of cutting it with a straight line such that its perimeter is halved. Can this be done if we specify the direction of the line?

What about if we specify a point to pass through?

Are there any points with more than one such line through them? If so, is this true of all points?

Is it possible to have exactly two such lines through any point?

I know it's a long problem, but any help with even a small part of it would be really really appreciated. Thanks in advance!!