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GoGeometry Action 113!

Creation of this applet was inspired by a tweet from Antonio Gutierrez (GoGeometry). Shown below is a triangle with its incircle drawn. The 3 smaller circles are tangent to the larger circle and are tangent to the sides of the triangle as well. The radii of these circles are shown as colored segments. Feel free to move the vertices of the triangle anywhere you'd like. How can we formally prove what is dynamically being illustrated here?

Quick (Silent) Demo