1/
a)
Ce qui suit en vecteurs.

AB.AC + BA.BC = AB.(AB+BC) + BA.BC
AB.AC + BA.BC = AB² + AB.BC + BA.BC
AB.AC + BA.BC = AB² + BC(AB + BA)
AB.AC + BA.BC = AB²
-----
b)
AB.AC + BA.BC = AB²
c.b.cos(alpha) + c.a.cos(beta) = c²
b.cos(alpha) + a.cos(beta) = c
-----
2/
a)
alpha + beta + delta = Pi   (la somme des angles d'un triangle = Pi)

Si beta = 2alpha ->
3.alpha + delta = Pi

delta = Pi - 3alpha

comme delta > 0 ->
Pi - 3alpha > 0
Pi > 3 alpha
3alpha < Pi
alpha < Pi/3
-----
b)

a/sin(alpha) = b/sin(beta) = c/sin(gamma)

sin(alpha) = (a/b).sin(beta)
sin²(alpha) = (a²/b²).sin²(beta)
cos²(alpha) = 1 - sin²(alpha) = 1 - (a²/b²).sin²(beta)
cos²(alpha) = 1 - sin²(alpha)
cos²(alpha)= 1 - (a²/b²).sin²(2.alpha)  
cos²(alpha)= 1 - 4(a²/b²).sin²(alpha).cos²(alpha)
1 - cos²(alpha)=  4(a²/b²).sin²(alpha).cos²(alpha)
sin²(alpha) = 4(a²/b²).sin²(alpha).cos²(alpha)
1 = 4(a²/b²).cos²(alpha)
cos²(alpha) = b²/(4a²)
cos(alpha) = b/(2a)

Il y avait plus court pour y arriver.    
-----
c)
c = a.cos(beta) + b.cos(alpha)
c = a.cos(2alpha) + b.b/(2a)
c = a.(2cos²(alpha)-1) + b²/(2a)
c = a.(2.b²/(4a²) - 1) + b²/(2a)
c = a.(2.b²-4a²)/(4a²) + b²/(2a)
c = (b²-2a²)/(2a) + b²/(2a)
c = (2b²-2a²)/(2a)
c = (b²-a²)/a  
b²-a² = ac
-----
Sauf distraction.