Ratio of triangles areas

In an equilateral triangle ABC, the point T is its centre of gravity, the point R is the image of the point T in axial symmetry, along the line AB, and the point N is the image of the point T in axial symmetry along the line BC. Find the ratio of the areas of the triangles ABC and TRN.

Correct result:

r =  3:1

Solution:

a=1 v=3/2 a=3/2 10.866  S1=a v2=1 0.86620.433 t1:t2=2:1 t1+t2=t=v  t2=13 v=13 0.8660.2887  TR=2 t2=2 0.28870.5774 TN=TR=0.57740.5774  BTN=60  cosBTN=v2:TN v2=TN cosBTN=TN cos60 =0.5774 cos60 =0.5774 0.5=0.28868  TN2=(RN/2)2+v22 RN=2 TN2v22=2 0.577420.28872=1  S2=RN v22=1 0.288720.1443  r=S1/S2=0.433/0.14433=3:1



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