Retexturing new triangles in the place of old ones

I have a simple situation to explain what I want to know. I am cutting a triangle into smaller triangles along a plane as so.

The result is the original textured triangle is now 3 triangles appropriately textured. One being the cut off corner, and the other a result of splitting the remaining trapezoid.

I vaguely understand that the texture coordinates are expressed by 0-1 values. I’m just not sure how I should be reassigning new values so that they remain the same. To find the intersection of the edges with the plane I get a t value per the 3 edges. I know that if cut, only 2 will be valid. That isn’t the issue here.

Thanks for your help.