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).

   |CT|     h*sqrt(2)         h         |CB|
  ------ = ----------- = ----------- = ------
   |CA|        2*h        h*sqrt(2)     |CT|

Therefore, triangles ACT and TCB are similar.

Therefore, angles CAT and CTB are equal.

Angle z is an exterior angle of triangle CBT.

Therefore, angle(z) = angle(CBT) + angle(CTB)

                    = angle(CBT) + angle(CAT)

                    = angle(y) + angle(x)

Posted by Bractals on 2005-12-24 04:35:23