Finding the length from an interior point of a triangle to a vertex given distances to the other two

Question

So let's assume that there is a triangle ABC and there is a point P inside of ABC. You are given the distances of AP and BP and you are trying to solve for CP. I faintly remember reading something like this with rectangles and a general formula in which you can determine the distance from the interior point to the fourth vertex given the distances to the previous three vertices. Is this true with an arbitrary triangle?

I have tried this with analytic geometry, assigning coordinate points to each vertex and using some basic Pythag and basic algebra to solve for the interior point (x,y) then simply using the distance formula to solve for the length to the last vertex. I have not come up with a general formula though. Would this only be possible if there were coordinates assigned to each point?

Answer 1

If all you know is the length of AP and the length of BP, then it is not possible to get the position of C.

Just think of a possible triangle with a point P inside and then move C. Although AP and BP (and even AB) do not change, CP changes.

Answer 2

Imagine two sticks of known lengths PA and PB where you can rotate one with respect to the other.(Right now A or B positions not fixed fully). It appears you have means of reaching C from earlier imagination with some constraint in case of a rectangle but not one clue to connect C is given here. How can you get to C that way?

So fully fix A and B along with angles or coordinates of A and B, next define relationship of these to C.