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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.

 Jul 30, 2021
 #1
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AB^2 = 18 - 6*sqrt(3)

AC^2 = 14 - 4*sqrt(3)

BC^2 = 20 - 8*sqrt(3)

 Sep 12, 2021

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