Twins of Arbelos and Circles of a Triangle

There are some Demonstrations dedicated to the remarkable twins of arbelos (orange disks in the graph above). This one involves them and an equilateral triangle of unit side (in blue) placed at the base of their semicircle (in gray). Consider points , , on the sides , , such that . Joining , and to the opposite vertex defines a triangle, of which you can choose to show either its incircle or its circumcircle (in green). By moving the slider for the red arrow you change the size of this circle. There are two positions in which the size of the blue circle and the size of the twins coincide—the purpose of this Demonstration is to show you the exact value when found.



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Problem proposed by Antreas Hatzipolakis.
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