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).

(In reply to only geometry? by Mindrod)

I'm not Paulo, obviously, but I want to comment on the meaning of "only geometry."  Paulo first submitted the problem with the restriction that algebra and trigonometry were NOT to be used.  When the problem came under review in the queue, Paulo changed his restriction to "only geometry" based on an objection by somebody (not me) to his original wording.  His intent clearly was to ban the use of  coordinates and trig identities, leaving us with only with what is called "synthetic geometry" to use -- that is, essentially Euclid's methods. And yes, cutting triangles out of paper and putting them together would be in the spirit of the intended approach, and perhaps "similar trangles" might somehow be found in this way that would lead to a good proof.

Posted by Richard on 2005-12-23 13:47:41