<Qua-2025-772>, trapezoid, similar 

<Romania District Olympiad 2010

Let T,T' be feet of altitudes DT, CT'. Since AB//CD triangles ADT, CBT' are congruent

Suppose AD=1, DE=t, CE=1-t, Since <CEF=<EAD, <ADE=<ECF, triangles ADE, ECF

are similar, so CF=EC x DE/AD=(1-t)t=-(t-1/2)^2+1/4=1/4 CB(=1)

when t=1/2, AD/CE=AE/EF=1/O.5, Hence triangles ADE, AEF are similar

<EAD=<FAE