[b]Suppose, in the applet below, [i]AB = a [/i]and [i]BC = b[/i]. [/b][br][br]Slide the slider slowly and carefully observe the dynamics that take place. [br]Be sure to move point [i]B[/i] around afterwards. [br]If you choose, drag [i]A [/i]and/or the large white point as well. [br][br]Reset the slider and re-slide slowly. [br][br][b]Write an expression, in terms of [i]a[/i] and [i]b[/i], for the length of the black segment (labelled with 3 hash marks.) [br][br][color=#0000ff]How can you formally prove your response true? [/color][/b]

Creation of this applet was inspired by a [url=https://twitter.com/gogeometry/status/849248130889641989]tweet[/url] from [url=https://twitter.com/gogeometry]Antonio Gutierrez[/url].