Inverse Pythagorean theorem

<table><tbody><tr><th>BasePytha-goreantriple</th><th><i>AC</i></th><th><i>BC</i></th><th><i>CD</i></th><th rowspan="6"></th><th><i>AB</i></th></tr><tr><td>(3,  4,  5)</td><td>20 =  4× 5</td><td>15 =  3× 5</td><td>12 =  3× 4</td><td>25 =  52</td></tr><tr><td>(5,       12,       13)</td><td>156 =       12×13</td><td>65 =  5×13</td><td>60 =  5×12</td><td>169 =       132</td></tr><tr><td>(8,       15,       17)</td><td>255 =       15×17</td><td>136 =  8×17</td><td>120 =  8×15</td><td>289 =       172</td></tr><tr><td>(7,       24,       25)</td><td>600 =       24×25</td><td>175 =  7×25</td><td>168 =  7×24</td><td>625 =       252</td></tr><tr><td>(20,       21,       29)</td><td>609 =       21×29</td><td>580 =       20×29</td><td>420 =       20×21</td><td>841 =       292</td></tr><tr><td colspan="6">All positive integer primitive inverse-Pythagorean triples having up to three digits, with the hypotenuse for comparison</td></tr></tbody></table>
<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows:
</p>
Let A, B be the endpoints of the <a href="/facts/Hypotenuse/hVMauM5f">hypotenuse</a> of a <a href="/facts/Right_triangle/dD6M9GAn">right triangle</a> △<i>ABC</i>. Let D be the foot of <a href="/facts/Altitude_(triangle)/z9H8fFMz">a perpendicular</a> dropped from C, the vertex of the <a href="/facts/Right_angle/psaslRq7">right angle</a>, to the hypotenuse. Then

1
            
              C
              
                D
                
                  2
                
              
            
          
        
        =
        
          
            1
            
              A
              
                C
                
                  2
                
              
            
          
        
        +
        
          
            1
            
              B
              
                C
                
                  2
                
              
            
          
        
        .
      
    
    {\displaystyle {\frac {1}{CD^{2}}}={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}.}

<p>This theorem should not be confused with proposition 48 in book 1 of <a href="/facts/Euclid/dHpVyL4R">Euclid</a>'s <i><a href="/facts/Euclid%2527s_Elements/nPKvCIxs">Elements</a></i>, the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">converse</a> of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle.
</p>

Inverse Pythagorean theorem open-in-new

Inverse Pythagorean theorem