The Golden Ratio: Equilateral Triangles

Here is the previous post concerning the Golden Ratio. For a collection of all the posts concerning the Golden Ratio, click #GoldenRatio below.

Sorry for the long wait for all those that were interested! Today we have something simple. We have seen that the golden ratio is related to triangles through the golden triangle and the Kepler triangle. But it is also related to the equilateral triangle. The above is an equilateral triangle drawn with its circumcircle. $X$ and $Y$ are on the line between the midpoints, $LM$ .

Prove that $\frac{LM}{MY} = \phi$

Credit: Cut-the-Knot

#Geometry #Construction #CosinesGroup #GoldenRatio

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$