Let me make sure I've got this.I don't understand "extended intersect" or "in K". I think what you are saying is that K is a point halfway between B and C and therefore along the same line as AQ if AQ was extended. Correct?
How do you know that QK=AQ/3. Is that a known law or is there a formula that tells you that. I have the same question for SO=SQ*3/4.
I've learned the theorem of cosines but I don't remember it off the top of my head. I've been known to get sine, cosine, cosecant and cotangent mixed up.