# Brachistochrone problem for 3 points

I wonder how I can solve the Brachistochrone problem for 3 points? The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in the system) I wonder what the shortest time path for that problem?

Is it enough to use Johann Bernoulli's solution from A to B and then Use the solution from B to C?

Or do I need to follow a different way to solve the shortest time path problem for 3 points?

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Yes, but one first has to generalize the classical 2-point Brachistochrone problem $A \to B$ where the initial speed $v_A$ traditionally is zero, to the case where the initial speed $v_A$ may be non-zero but fixed. The solution to this initial speed Brachistochrone problem (assuming no friction) is still a cycloid.
Now consider the 3-points Brachistochrone problem $A \to B\to C$ with initial speed $v_A$. The speed $v_B$ is given by energy conservation alone. Thus the two segments $A \to B$ and $B \to C$ are completely decoupled, and they can be optimized as two independent 2-point Brachistochrone problems with initial speeds $v_A$ and $v_B$, respectively, leading to two corresponding cycloids $A \to B$ and $B \to C$.