We proceed by proving the following important result
Theorem: (Sine Rule) In a triangle where and is the radius of circumcircle.
Similarly in we have . Hence the proof.
Now for the similarity theorems we have start by assuming the AAA similarity postulate as the definition and prove SSS, SAS or assume SSS similarity postulate and prove AAA, SAS.
Here I assume SSS postulate and prove the other two theorems
AAA: Two triangles such that are similar.
From Sine Rule we have
Similarly hence by SSS we have
SAS: The argument is same as above use sine rule to get the sine angles are equal and argue that the angles are themselves equal and not supplementary. And use AAA.