Look at the drawing below, where AB = BC = CD = DT = h, and TD perpendicular to AD:
o T
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+----------+----------+----------o
A (h) B (h) C (h) DThe 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).
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.
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Posted by goFish
on 2005-12-24 04:44:58 |
Complex solution
