Incenter is the point of intersection of the angle bisectors of a triangle. Let's call the incenter of triangle ABC P, and let's call the point where the incenter intersects side AB as D, and where the incenter intersects side AC as E.
Since incenter is the angle bisector, it is equidistant from all sides of the triangle. Therefore, PF is the incenter's distance to side BC.
We can use the angle bisector theorem to help us solve this problem. The angle bisector theorem states that in a triangle, an angle bisector divides the opposite side into two segments that are proportional to the other two sides of the triangle.
So, we have the following proportion:
|AB|/|AD| = |AC|/|AE|
Now we can substitute the given values:
|AB|/(2x-5) = |AC|/(3x-15)
Then, we can use the fact that PF is the incenter's distance to side BC and is equal to the proportion of |AB| and |AC| (since we assume that |AD| = 2x-5 and |AE| = 3x-15). Therefore:
PF = |AB| + |AC| = 2x-5 + 3x-15 = 5x-20
So, the distance PF is 5x-20.
