Basicly you should do as little cacluations as possibly so:

1. You know in what triangle your player is (or is moving onto)
2. You know if the triangle is potentionaly collidable or not
3. If it is not collideable the player is happy and can stay ther (or move to)
4. If it is potentionaly collideable we need to investigate further
5. You know your player sprite, with an transparency colour and its x,y pos and size
6. You know your triangle sprite (background texture) with an transparency position and x,y and size
7. now first you check if their boundaries collide (most likely) if not the player is happy and can stay there (or move to)
8. if it collides you can invstigate further by pixel comparing the colliding boundary you got in step 7. As long as you only get transparent matches your player is happy, but if one pixel turns up to be nontransparent you realy have an collision and your player is unhappy.

PS could you make a posting on how i should visualize a triangulated map? I gues it is something like a tilemap but not with squares but triangles instead?
PS2 i assume the player is not larger then the triangle.