Intersecting secants theorem

In <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, the intersecting secants theorem or just secant theorem describes the relation of <a href="/facts/Line_segment/V8E0M4qk">line segments</a> created by two intersecting <a href="/facts/Secant_line/gJnaF5g3">secants</a> and the associated <a href="/facts/Circle/9fy5ggKn">circle</a>.
For two lines AD and BC that <a href="/facts/Line%25E2%2580%2593line_intersection/41R6tjCG">intersect each other</a> at P and for which A, B, C, D all lie on the same circle, the following equation holds:

 
 
 
 
 |
 
 P
 A
 
 |
 
 ⋅
 
 |
 
 P
 D
 
 |
 
 =
 
 |
 
 P
 B
 
 |
 
 ⋅
 
 |
 
 P
 C
 
 |
 
 
 
 {\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}

The theorem follows directly from the fact that the <a href="/facts/Triangle/cRXUwsMn">triangles</a> △PAC and △PBD are <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a>. They share ∠DPC and ∠ADB = ∠ACB as they are <a href="/facts/Inscribed_angle/fTYeVKxt">inscribed angles</a> over AB. The similarity yields an equation for <a href="/facts/Ratio/mFmhQt7T">ratios</a> which is equivalent to the equation of the theorem given above:

P
              A
            
            
              P
              C
            
          
        
        =
        
          
            
              P
              B
            
            
              P
              D
            
          
        
        ⇔
        
          |
        
        P
        A
        
          |
        
        ⋅
        
          |
        
        P
        D
        
          |
        
        =
        
          |
        
        P
        B
        
          |
        
        ⋅
        
          |
        
        P
        C
        
          |
        
      
    
    {\displaystyle {\frac {PA}{PC}}={\frac {PB}{PD}}\Leftrightarrow |PA|\cdot |PD|=|PB|\cdot |PC|}

Next to the <a href="/facts/Intersecting_chords_theorem/FGAhKIrv">intersecting chords theorem</a> and the <a href="/facts/Tangent-secant_theorem/LIH2DEeq">tangent-secant theorem</a>, the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the <a href="/facts/Power_of_a_point/KVt6VnD6">power of point theorem</a>.

<ul><li>S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9789401169820, pp. <a href="https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA175">175-176</a></li>
<li>Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780470591796, p. <a href="https://books.google.com/books?id=Ch5CrMtyniEC&pg=PA161">161</a></li>
<li>Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-411-04208-1, pp. 415-417 (German)</li></ul>

Intersecting secants theorem open-in-new

Intersecting secants theorem