Geometry: move of "x" centimeter on a curve

Definition:
I have two connected line segments (in black on picture) defined by 3 points: P1 P2 and P2 P3.I subdivise first 5 centimeters of line segment (in grey on picture).
I connect corresponding subdivisions (in blue on picture).
If I subdivide again the first 5 centimeters at infinite, I will get a perfect curve (in red on picture) formed by connection lines.

Problem:
I'm on point P1 and I want to move of "x" centimeter toward P3 passing by line segment P1 P2, then on red curve and finally on line segment P2 P3.
How compute the exact new point coordinate of this move ?

Solution:
I only have an approximate solution where I compute the blue lines intersection points (in orange on picture).
Then, I compute length between orange points.
Finally, I move from P1 to 4nd subdivision of line segment P1 P2, then to first orange point, then on second orange point, etc. until I reach the "x" centimeter.