So before I can help I need to know how much you know about vectors.
For example, do you know what a dot product is?

Thankyou

sfwc
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The easy way is by using what is called the Steiner Theorem in classical Mechanics. The theorem establishes,
given N mass points, the Moment of Inertia to a particular point M (what means the sum of the product of each individual mass time the squared distance from each point to M) is equal to the Moment of Inertia of the system of point to its center of gravity G plus the product of the total mass of the N points times the squared distance GM.

So your triangle is a particular case of three masses.
Sure you will find a book where the theorem be better explained but if it is not the case see below

It is very easy to prove just take two perpendicular axis passing by G as origin, or three if you work in three dimension, call the coordinates of M, xm , ym, zm, then call the coordinates each vertex of your triangle xa , ya, za, ; xb , yb, zb , ; xc , yc, zc

So the Moment of Inertia of our system of three point referred to the point M will be :

Ma (xa –xm)2 +Ma (ya –ym)2+Ma (za –zm)2 = Maxa2 + Maxm2 - 2Maxaxm + May2a +…..

Mb (xb –xm)2 +Mb (yb –ym)2+Mb (zb –zm)2 = Mbxb2 + Mbxm2 - 2Mbxbxm + ay2b+…..

Mc (xc –xm)2 +Mc (yc –ym)2+Mc (zc –zm)2 = Mcxc2 + Mcxm2 - 2Mcxcxm + May2c+…..

Summing and grouping we will have :

Ma (xa –xm)2 +Ma (ya –ym)2+Ma (za –zm)2 + Mb ( xb –xm)2 +Mb (yb –ym)2+Mb (zb –zm)2 + Mc (xc –xm)2 +Mc (yc –ym)2+Mc (zc –zm )2 =

Maxa2 + Mbxb2+ Mcxc2+ xm2(Ma +Mb +Mc) - 2axm (Maxa+ Mbxb +Mcxc) +….

The term underline must be zero by definition of center of gravity
Using your notation we make: Ma ,Mb, Mc =1; and :xm2 + ym2 + zm2 = MG2

(xa2+ ya2+ za2) = AG2 ; (xb2+ yb2+ zb2) = BG2; (xc2+ yc2+ zc2) = CG2

(xa –xm)2 + (ya –ym)2 + (za –zm)2= MA2;
(xb –xm)2 + (yb –ym)2 + (zb –zm)2= MB2;
(xc –xm)2 + (yc –ym)2 + (zc –zm)2 = MC2;

Now we have MA2+ MB2+ MC2= (1+1+1) MG2+ AG2+ BG2+ CG2