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No trigonometry, no algebra (Posted on 2005-12-23) Difficulty: 3 of 5
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).

See The Solution Submitted by pcbouhid    
Rating: 3.0000 (1 votes)

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Complex solution | Comment 7 of 11 |

Being in festive mood and ignoring the handcuffs in the question:

If we let AD be the real line and represent the position of E relative to A,B and C by the complex numbers 3 + i, 2+ i and 1+i, then their product is 10 i. Since Arg(3 + i)=x, Arg(2 + i)=y, Arg(1 + i)=z, this shows that x+y+z = 90 degrees. Since z = 45, x+y = 45 = z.

Geometrically, what I am doing is making copies of BDE and ADE, and putting A and B at C.  Then I align BD with CE, align AD with the new BE. Then I show that the new AE is perpendicular to CD.


  Posted by goFish on 2005-12-24 04:44:58
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