Semicircle and similar triangles

Question

In a semicircle with center O and diameter AB take a point C between A and O. Erect a perpendicular at C striking semicircle at point E. Choose any point D on semicircle between A and E. Point F on semicircle is between E and B such that angle DCE equals angle ECF. Prove triangle DCE is similar to triangle ECF!

Answer 1

Let $\Phi$ be our circle, $DC\cap\Phi=\{D,K\}$ and $EC\cap\Phi=\{E,M\}$.

Since $\measuredangle ECF=\measuredangle MCK$ and the circle is symmetrical in relation to $AB$,

we obtain that $CF$ and $CK$ they are symmetrical in relation to $AB$, which gives that $$\smallsmile EF=\smallsmile MK,$$ which gives $$\smallsmile FBKM=\smallsmile EFBK,$$ which says that $$\measuredangle CEF=\measuredangle CDE$$ and we are done!