Construct a circle of any radius and let chords AB and CD be parallel
Join AD, BC and let E be the intersection of these chords
Now....because AB and CD are parallel and AD and BC are transversals, then angle BAE = angle CDE
And angle ABE = angle DCE for the same reason
And angle ABE = angle CDE because they intercept the same arc
And angle BAE = angle DCE for the same reason
Therefore....angle CDE = angle DCE...so....in triangle CED.....CE = DE
And....angle BAE = angle ABE....so..in triangle AED.....AE = BE
Therefore....AE + DE = CE + BE
And....AE + DE = AD
And CE + BE = BC
So....AD = BC