66 PRACTICAL MATHEMATICS 



If A is the area of the triangle, 



A = Vs(s - P!)(S - p z )(s - p 3 ) 



where s = - (p t 



If h lf h z , and Ii 3 are the lengths of the perpendiculars drawn 

 from the points P x , P 2 , and P 3 to the opposite sides respectively, 



1 2A 

 then A = - p^h-, or ft, = 



2 Pi 



1 2A 



/\ _ _ ff\ fa /"\T* n 



* ~" \J o't'O wl /to 



P2 



A 1 \ 7 2A 



A = - r>,w 3 or /?3 = 



FIG. 21. 



If X , 2 , and 3 are the angles of the triangle, 



1 

 then A = - p 2 p- sin 0, or sin 



2A 



aPs 



2A 



A D Q 



A = - ^gjjj sm 2 or sin = 



A 1 -n n 2A 



A = - 79,790 sin Oo or sin bo = - 



2 



and from these relations the angle between two given lines can 

 be determined. 



Also the angles lf 2 , 3 can be found by means of the following 

 relations : p 2 + p 2 _ p 2 

 cos 0, = 2 2 L 



cos 9 = 



COS 00 = 



