Showing that some curve is a circle

[Werg22]

I'm trying out some exercises in differential geoemtry and came across this one:

If \gamma(t) is unit-speed, and that all its normals pass through a given point, show that the trace of \gamma(t) is part of a circle.

My solution so far:

Let a be a point where all the normals pass, then we know that \gamma(t) - a = s(t) N(t) \ \forall t, where s(t) is a real-valued function.

But where should I take it from there? All I know is that ultimately I want to show that |s(t)| is constant for all t.

 

[zhentil]

You might as well assume that you start at (1,0) with initial velocity (0,1), and that all your normals pass through zero. You get a system of ODEs with a very simple solution :)