To get a better idea of the community's current geometric proof "skill level," I will keep dropping the difficulty of these problems until we find an adequate starting point for the synthetic geometry group. I will appreciate any feedback on today's problem as you approach it.
Given \(\triangle ABC\), let \(D,E\) denote the midpoints of the arc \(BC\), where \(A,E\) lie on the same side \(BC\). Construct \(F\) on \(AB\) such that \(CF\perp AB\), and \(G\) on \(AE\) such that \(GF\perp DF\). Prove that \(CG=EC\).