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Could you help with a Euclidean plane geometry problem?
If WXYZ is a rectangle, U is on XY and V is on YZ. We know that the following triangles are of equal area: triangle WXU, triangle UYV, triangle VZW. If a=XU, b=UY, c=YV, d=VZ, then prove that b/a will be the same as c/d and that b/a is the golden ratio.

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(1) Do you mean that the areas of the triangles are equal, or that the triangles are congruent? (2) If $V$ is on $YZ$, there is no triangle $VYZ$. –  Brian M. Scott May 13 '12 at 21:48
    
You don't happen to have a nice diagram on you, do you? –  J. M. May 13 '12 at 22:04
    
J.M. I would like to upload a diagram, but I'm not sure how to draw it on this site. I would not mind drawing free hand. –  user31284 May 13 '12 at 22:29
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You can upload any pic (made with e.g. mspaint on windows) to imgur and provide the url here; someone with sufficient rep (I forgot what the threshold is) will swoop in and put it into your post. –  anon May 13 '12 at 22:38
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You should be able to upload a picture here. Sixth button from the left of the panel above the text box, there is an image uploader. Follow the instructions. –  J. M. May 13 '12 at 22:39

1 Answer 1

up vote 2 down vote accepted

The area of triangle $WXU$ is $\frac12a(c+d)$; that of triangle $UYV$ is $\frac12bc$; and that of triangle $VZW$ is $\frac12d(a+b)$. If these areas are equal, we have $$\frac12a(c+d)=\frac12bc=\frac12d(a+b)$$ and hence $ac+ad=bc=ad+bd$.

The first part of the answer now follows very easily from the fact that $ac+ad=ad+bd$; just do a little algebra.

Once you’ve shown the first part, divide the equation $ac+ad=bc$ by $bd$ to get $$\frac{a}b\left(\frac{c}d+1\right)=\frac{c}d\;,$$ let $x=\dfrac{b}a$, and solve for $x$ to complete the problem.

Here’s a diagram, not to scale:

enter image description here

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The areas of the triangles are all congruent. I can see how that reads ambiguously. I appreciate your help. –  user31284 May 14 '12 at 3:26
    
@user31284: It makes no sense to say that the areas are congruent: congruence is a property of shapes, and areas are numbers. It seems that what you meant is simply that the areas are equal. –  Brian M. Scott May 14 '12 at 3:28
    
My mistake. I will edit this. –  user31284 May 14 '12 at 5:12
    
@user31284: That’s the percentage of your questions for which you have accepted an answer. See this part of the FAQ to learn how to accept an answer. –  Brian M. Scott May 17 '12 at 3:31

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