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GoGeometry Action 18!

Suppose, in the applet below, AB = a and BC = b. Slide the slider slowly and carefully observe the dynamics that take place. Be sure to move point B around afterwards. If you choose, drag A and/or the large white point as well. Reset the slider and re-slide slowly. Write an expression, in terms of a and b, for the length of the black segment (labelled with 3 hash marks.) How can you formally prove your response true?
Creation of this applet was inspired by a tweet from Antonio Gutierrez.