In $a$ triangle ABC, $a$ point P in the interior of ABC is such that ∠BPC − ∠BAC = ∠CPA − ∠CBA = ∠APB − ∠ACB . Suppose ∠BPC = 30° and AP = 12. Let D, E, F be the feet of perpendiculars form P on to BC, CA, AB respectively. If $mn$ in the area of the triangle DEF where m, $n$ are integers with $n$ prime, then what is the value of the product mn?

Step 1:  2 3 

Step 2: EF 2 = l 2 cos2 (  − 30° ) + cos2 ( 60° −  ) − cos (  − 30° ) cos (  − 60° )

Step 3:  2 

Step 4:  3 1

Step 5: 1 3

Step 6: 3 3 

Step 7: = l 2  cos  + sin  + cos  + sin  − cos ( 2 − 90° ) + 

Step 8:  2 2 2

Step 9: 2

Step 10: 2 2 

Step 11:  

Step 12:  3 3

Step 13: = l 2 cos2  + sin2  + sin  cos  − 

Step 14:  2 4 

Step 15: l2 3 3 144

Step 16: EF 2 = , area = .EF 2 = =9 3

Step 17: 4 4 4 4