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A golden triangle is a triangle ABC where the sides AB and AC are of equal length and their length is greater than the length of side BC. Also the ratio of the length of the side AB to the length of the side BC is equal to [1 + sqrt 5] / 2.
So a golden traingle is isosceles, the length of the sides that are equal is greater than the length of the third side and the ratio of the length of the equal sides and the third side is [1 + sqrt 5] / 2.
A golden ratio is (1+sqrt5)/2. It is the positive solution of of the equation x^2-x-1 = 0.
A golden triangle is a special isosceles triangle whose isosceles side and the base are in the ratio (1+sqrt5)/2 : 1. Or (1+sqrt5):2.
The angles of the golden triangles are 72 degree , 72 degree and 36 degree. Or the angles of a golden triangle are in the ratio 2:2:1.
In a regular 10 sided regular polygon, the the lines joining the centre and the ends of a side make the required golden triagle.
We can construct the golden triangle very easily.
We can our own unit AB., drawn on a paper.
Take 2 units and 3 units as the side of a right angle and its hypotenuse. Then the other side gives us the value of sqrt(3^2-2^2) = sqrt5. So with our own unit , sqrt 5 units we can construct a special golden triangle of 2 units base and each of isosceles sides with (sqrt5+1) units.
We consider a gold acute triangle, ABC, with the base length AB=l and the other 2 sides with the lengths g, where g is the gold ratio and m is the measure of the equal angles of the ABC triangle.
We consider a point D set on the side BC so as AD=l. The line AD split the triangle in 2 smaller triangles: ABD and ADC. Because AD=AB=l => the measure of the angle ABD = the measure of the angle ADB=m => the triangle ABD is a similar triangle with the ABCtriangle => BD=g-l => CD=l.
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