Let ABCD be a quadrilateral where AB = BC, CD = 2AB, m(∠ABC) = m(∠BCD) = 120°.
a) Prove that ∆ABD is isosceles.
b) Calculate the values of the trigonometric functions of the angle ADB.
c) Prove that the points A, M, N are collinear, where M and N are the means of the sides BD and CD, respectively.
d) Prove that the line CD is tangent to the circumscribed circle of the triangle AMD.
e) Find the area of the quadrilateral ABCD, depending on AB = a.
f) Calculate the sine of the angles formed by the lines AC and BD.