property of inscribed angles 
  
    
      
        ⇒
      
    
    {\displaystyle \Rightarrow }
  
 
  
    
      
        ∠
        P
        
          G
          
            2
          
        
        T
        =
        ∠
        P
        T
        
          G
          
            1
          
        
      
    
    {\displaystyle \angle PG_{2}T=\angle PTG_{1))
  
 
  
    
      
        ⇒
      
    
    {\displaystyle \Rightarrow }
  
 
  
    
      
        △
        P
        T
        
          G
          
            2
          
        
        ∼
        △
        P
        
          G
          
            1
          
        
        T
      
    
    {\displaystyle \triangle PTG_{2}\sim \triangle PG_{1}T}
  

  
    
      
        ⇒
      
    
    {\displaystyle \Rightarrow }
  

  
    
      
        
          
            
              
                |
              
              P
              T
              
                |
              
            
            
              
                |
              
              P
              
                G
                
                  2
                
              
              
                |
              
            
          
        
        =
        
          
            
              
                |
              
              P
              
                G
                
                  1
                
              
              
                |
              
            
            
              
                |
              
              P
              T
              
                |
              
            
          
        
      
    
    {\displaystyle {\frac {|PT|}{|PG_{2}|))={\frac {|PG_{1}|}{|PT|))}
  

  
    
      
        ⇒
      
    
    {\displaystyle \Rightarrow }
  

  
    
      
        
          |
        
        P
        T
        
          
            |
          
          
            2
          
        
        =
        
          |
        
        P
        
          G
          
            1
          
        
        
          |
        
        ⋅
        
          |
        
        P
        
          G
          
            2
          
        
        
          |
        
      
    
    {\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}

The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Elements.

Given a secant g intersecting the circle at points G1 and G2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds:

The tangent-secant theorem can be proven using similar triangles (see graphic).

Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.

References