Here is a rather quirky derivation that doesn't use calculus notation. I believe that the general idea dates back to Huygens (c 1659).

A small arc of a circle is almost indistinguishable from part of a parabola. You can show using Pythagoras's theorem that if $x$ is the semi-length of the chord that runs between the ends of an arc of a circle of radius $r$, and $h$ is the sagitta of the arc (the distance between the centre of the arc and the centre of the chord) then
$$x^2=2rh\ \ \ \ \ \ \ \ \text {provided $h$ is small enough to neglect $h^2$.}$$
In order to follow the path, a body at the midpoint of the arc at $t=0$ and moving parallel to the chord at speed $u$ must have an acceleration $a=\frac {u^2}r$ at right angles to the chord. We know this from ordinary projectile theory... At time $t$ from passing the centre of the arc,
$$x=ut\ \ \ \ \ \text{and}\ \ \ \ \ h=\tfrac 12 at^2\ \ \ \ \ \text{so}\ \ \ \ \ \ x^2=2\left(\frac{u^2}a\right) h$$
Comparing with our first equation, $r=\frac{u^2}a$, that is $a=\frac {u^2}r$.