The following polar grid is centered at the origin, with concentric circles of positive integer radii starting at $1,$ and consecutive rays through the origin with angles of pi/12 between them:
[asy] size(200); pair A = (0,0); for (int i = 1; i < 6; ++i) { draw(Circle((0,0),i), linewidth(0.4)); } for(int i=0;i<360;i+=15) { draw(rotate(i)*((-5.0,0)--(5.0,0)), linewidth(0.4)); } pair A, B, C; A = 4*(1/2, sqrt(3)/2)/dir(15); B = 3*dir(45)*(sqrt(3)/2, 1/2); C= 2*(1/2, sqrt(3)/2)/dir(45); dot(A, red+linewidth(3.5)); dot(B, red+linewidth(3.5)); dot(C, red+linewidth(3.5)); label("$A$", A,N); label("$B$", B, NNW); label("$C$", C, S); [/asy]
For the points $A, B$ and $C$ above, figure out AB^2, AC^2, BC^2and enter them in that order below.
+132 I got this answer instead using the law of cosine: \[AB^2=25-12\sqrt{3}, AC^2= 20-8\sqrt{3},\] and \[BC^2= 7\]
