Consider a right triangle with an inscribed circle. Let x and y be the lengths of the two line segments formed on the hypotenuse by the point of tangency with the circle. What interesting fact can you prove about x*y?

   
    A     r      E             y                  D
     * '''''''''''' * '''''''''''''''''''''''''''''''''''''''''' *
     |            |                                  |
x   |            * F                               |
     |            |      * T                       |
     |            |                 H               |
     * '''''''''''' * ''''''''''''''''''''' * '''''''''''''''' * G 
r    |           O                                  |
     * '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''*
     C                                              B   

The given right triangle is ACB, the center of the incircle is O.  The incircle is tangent to the hypotenuse at T. We wish to demonstrate that the area x*y of the rectangle EOGD is equal to the area of  ACB. Observe that EOGD - FTO - HTO + AEF + HGB = ADB.  But AEF = FTO and HTO = HGB. Hence  area EODG = area ADB.  But ADB = ACB and thus we have area EODG = area ACB as desired.

 

Edited on March 9, 2005, 12:36 am

Posted by Richard on 2005-03-01 00:48:56