Problems

The triangle \(BCD\) is inscribed in a circle with the centre \(A\). The point \(E\) is chosen as the midpoint of the arc \(CD\) which does not contain \(B\), the point \(F\) is the centre of the circle inscribed into \(BCD\). Prove that \(EC = EF = ED\).

Let \(A\), \(B\), \(C\), \(D\), \(E\) be five different points on the circumference of a circle in that (cyclic) order. Let \(F\) be the intersection of chords \(BD\) and \(CE\). Show that if \(AB=AE=AF\) then lines \(AF\) and \(CD\) are perpendicular.