Top: The golden ratio φ is obtained by taking the distance from the midpoint of the lower side of the square to the vertex on the lower extension.
Bottom: If isosceles triangle A with base 1 and isosceles φ is placed inside isosceles triangle B with base 1/φ and isosceles 1, which is similar to itself, the remaining part C becomes an isosceles triangle with base φ and isosceles 1. It follows from this that the magnitude of the interior angles of the triangle are 1:2:3. This triangle is similar to the triangle that appears when a pentagram is superimposed on a regular pentagon.
Middle: If a line segment is divided into φ:1:φ and its φ+1 part is further divided into φ:1:φ, the left endpoint of the earlier division coincides with the right endpoint of the later division. ((φ+1)/(2φ+1)*(φ+1)=φ) The algorithm that uses this feature is the golden partition search.
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