Proof that the tautochrone is a cycloid In the Wikipedia article about the tautochrone curve, there is a proof of the fact that the tautochrone curve must be a cycloid. The proof starts with the following statement:

One way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator: the height of the curve must be proportional to the arclength squared.

How is the statement in bold justified?

The statement in bold is better phrased as:

If $s(t)$ describes simple harmonic motion, then the curve traced by the particle will be an isochrone.

The motivation for this assumption is the fact that in simple harmonic motion, the period is independent of the amplitude of the motion. So if $s(t)$ could be arranged to follow SHM, the particle's time of descent to its lowest level would be the same regardless of its starting position -- which is the goal of a tautochrone.
To understand what's going on in the rest of that paragraph, recall that the Lagrangian under the SHM assumption would then be
$$
L(s) = \text{kinetic energy - potential energy} = \frac12 m\dot s^2 -\frac12 ks^2.
$$
This explains why the paragraph is supposing that $y=s^2$ ("the height of the curve must be proportional to the arclength squared"), since the particle's potential energy is proportional to its vertical displacement above the lowest level, which is what $y$ represents. The rest of that paragraph goes on to derive the cycloid solution from the assumption $y=s^2$. 
