An incredible semicircle problem!

A semicircle contains an inscribed semicircle dividing its diameter into two lengths a and b. Can you find the formula for the inscribed semicircle's diameter in terms of the lengths a and b? What is the locus of the center of the inscribed semicircle? Thanks to Nick from Greece for the suggestion! Special thanks this month to: Mike Robertson, Michael Anvari, Daniel Lewis, Robert Zarnke, Kyle. Thanks to all supporters on Patreon! / mindyourdecisions
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