Problem Study (Geometry applying Trigonometry and Calculus)

(a)    Let be a triangle with right angle and hypotenuse .  
        (See the figure.) 
        If the inscribed circle touches the hypotenuse at D, 
        show that .
(b)   If  , express the radius of the inscribed circle in terms  of and
(c)   If  is fixed and  varies, find the maximum value of .  
 Solution       
    Let   be the centre of the circle and and be points of            
    tangency of and respectively.        
   (given)        
   Draw and       
          
   Since and , is a square.
      Therefore         
      Let .      
          
(a)            
              
             
             
       
(b)   Draw . Since is incentre, bisects .        
        In right ,         
        
       In right , By Pythagoras Theorem,
               
               
      
       
     
        
       
       
       
     
     
     
       
     
     
       
(c)   Since is fixed and  varies, is a function of and the rate of 
       change of with respect to is
        
       
      has stationary value when  .
       
       
       
        and .  
        
              
      When ,
      .
      will be maximum value when .