An equilateral triangle and several line segments. What fraction is purple?

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## Solution

The purple fraction is 1/7.

## Poem

The holy number

It could be 777

Straight across and right from heaven

Or maybe another one

But it could be a number

The golden ratio

That’s not even a 0

Known from Pythagoras and Euclide

The golden number in geometry

A number that’s holy

For thinkers, poets in each century

## One reply on “The holy number”

ABC equilateral triangle with AB top horizontal side. D midpoint of AB. E midpoint of DC. F is point on AE and DF perpendicular to AE.

Triangle DFB = triangle ADF.

AD=1, DE=β3/2. Hence AE=β7/2

Triangle ADE = (1/2)*AD*DE = (1/2)*DF*AE giving DF=β3/β7

Hence AF=2/β7 from triangle ADF

Area of triangle ADF=(1/2)*AF*DF=β3/7

Area of triangle ABC=(1/2)*AD*DC=β3

Hence the proportion is (β3/7)/(β3)=1/7 or 0.142857