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# The glass pyramid

The vertices of an equilateral triangle are connected to an interior point. What is the angle α?

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## Solution

The angle α is 45°.

Some elementary angle chasing gives the above. Now we can apply the Law of sines in the bottom triangle: sin(α)sin(3α)=sin(210-4α). This can be rewritten using trigonometric rules as (cos(2α)-cos(4α))/2=sin(4α)√3/2-cos(4α)/2, which simplifies to cos(2α)=sin(4α)√3=cos(2α)sin(2α)2√3.

So, either cos(2α)=0 or sin(2α)=√3/6. The latter solution does not give an interior meeting point. The former amounts to α=45°.

## Poem

The glass pyramid
A sublime monument
From ancient times
Shines like a diamond
Amidst the neon lights
A pyramid of glass, metal or stone
The moon is reflected
In its glittering façades
And its countless steps
Gigantic stairway for the soul
To ascend to heaven
The final goal

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## One reply on “The glass pyramid”

Benjamin Tzursays:

My evaluation no one answer ;
Any 52.5》alpha 》33.5