Geodesic curvature

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Homework Statement


Show that the geodesic curvature of an oriented curve C in S at a point p in C is equal to the curvature of the plane curve obtained by projecting C onto the tangent plane along the normal to the surface at p.


Homework Equations


Meusnier's theorem, and k^2 = (k_g)^2 + (k_n)^2


The Attempt at a Solution


The proposition makes sense. It's basically saying that the geodesic curvature is what's left after you take out the effect of how the surface is curved in the ambient space. But how to prove it?
I used Meusnier's theorem to get that the normal curvatures of both curves are the same. But I don't know what to do from there. Any help?
 

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