Look at the drawing below, where AB = BC = CD = DT = h, and TD perpendicular to AD:

                                   
                                   o T
                                   |
                                   | (h) 
                                   |
                                   |
  +----------+----------+----------o 
  A    (h)   B    (h)   C    (h)   D

The line AT makes an angle x with AD; the line BT makes an angle y with BD; the line CT makes an angle z with CD.

Using only geometry, prove that (angle x) + (angle y) = (angle z).

                                      1     1
                                     --- + --- 
               tan(x) + tan(y)        3     2
  tan(x+y) = ------------------- = ------------- = 1 = tan(z)
              1 - tan(x)*tan(y)          1   1
                                    1 - ---x---
                                         3   2

Therefore, z = x+y

 

Posted by Bractals on 2005-12-24 03:41:34