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.
Golden Ratio within quadrilateral
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geometry
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0@Gerry, he's got 16 rep, which is sufficient if memory serves... – 2012-05-14
1 Answers
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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:
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0@user31284: That’s the percentage of your questions for which you have accepted an answer. See [this part of the FAQ](http://math.stackexchange.com/faq#howtoask) to learn how to accept an answer. – 2012-05-17