Hello, billionj!
Determine the length of the two longest sides on an isosceles triangle
with the angles of 36°, 72°, 72°, and the base length of 550 feet. This is probably not an acceptable solution to this problem.
But it is a fascinating bit of mathematical trivia . . . hope you enjoy it.
Code:
*
/ \
/36°\
/ \
/ \ φ
/ \
/ \
/ 72° 72° \
* - - - - - - - *
1
This is a very special isosceles triangle.
The ratio of the equal side to the base is: . 
. . . the Golden Ratio.
Hence, with this triangle:
Code:
*
/ \
/36°\
/ \
x / \
/ \
/ \
/ 72° 72° \
* - - - - - - - *
550 we have: . 
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The Fibonacci sequence: . 
. . where each term is the sum of the preceding two terms.
The ratios of consecutive terms approach the Golden Mean.
. . ![\begin{array}{ccc}<br />
\phi & = & 1.618 033 989 \\ \hline \\[-3mm]<br />
\dfrac{55}{34} &=& 1.617 647 059 \\ \\[-2mm]<br />
\dfrac{233}{144} &=& 1.618 025 751 \\ \\[-2mm]<br />
\dfrac{987}{610} &=& 1.618 032 787 \\ \\[-2mm]<br />
\dfrac{2584}{1597} &=& 1.618 033 813 \\ <br />
\vdots && \vdots \end{array}](http://latex.codecogs.com/png.latex?\begin{array}{ccc}<br />
\phi & = & 1.618 033 989 \\ \hline \\[-3mm]<br />
\dfrac{55}{34} &=& 1.617 647 059 \\ \\[-2mm]<br />
\dfrac{233}{144} &=& 1.618 025 751 \\ \\[-2mm]<br />
\dfrac{987}{610} &=& 1.618 032 787 \\ \\[-2mm]<br />
\dfrac{2584}{1597} &=& 1.618 033 813 \\ <br />
\vdots && \vdots \end{array})
Hence, any pair of consecutive terms of the Fibonacci sequence
. . will form a triangle very close to the 36°-72°-72° triangle.
Example:
Code:
A
*
/ \
/ \
/ \
377 / \ 377
/ \
/ \
/ \
B * - - - - - - - * C
233
The vertex angle is: . 
The base angles are: . 
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Originally Posted by billionj 
