Two isosceles triangles are placed as shown. What is the proportion of the line segments a and b?
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The proportion a : b is 1 : 2.
Now compare right triangles AFB and CEB. They have the same angle α en hence they are similar.
Now from the 30-60-90 triangle BFE it is seen that BF : BE = 1 : 2. Therefore also the hypotenuses a : b = 1 : 2.
Note 1: triangle ABC also is a 30-60-90 triangle; one of the many in this interesting figure.
Note 2: all possible configurations are dynamically shown in this beautiful animation by Ignacio Larrosa Cañestro.