Ratio Formula: (Internal Division)

            Let and be any two points on the line.
Let a point be the point which divides PQ in the ratio  i.e.
                        PR : PQ =
From P, R and Q draw PM, RN and OL perpendicular to x-axis.
From 'P' draw to RN.
From 'R' draw to QL.
Since the right triangles
   and
 
  
   
       and  are similar triangle.
                                                    
                                                    
                                                    
                                                    
                                                   
                                       I.    
                                                Cross Multiplication
                                                    
                                                    
                                                    
                                                    
                                                   
                                         II.   
                                                Cross Multiplication
                                                 
                                                  
                                                  
                                                
                                               
III.                                           Required Point
                                  
Ratio Formula for External Division:

            Let R(x, y) be the point which divides PQ externally
then                                
so                                    K1:K2 = K1: -K2
we have ratio formula
               
then
               
               

Mid Point Formula:
We have ratio formula
                               
If P(x, y) is the mid point of the segment QR, then
               

So ratio formula becomes
                               
                               
                               

                                    OR