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Picture proof of trigonometric angle-sum formulas

You can drag points B, C, and D. Scroll down for more explanation. Based on an image by Tamás Görbe (twitter @TamasGorbe)

How to construct (and why it works):

1. Start with the bottom right triangle (ABC) 2. Draw the right triangle ACD with one leg on the hypoteneuse of ABC. 3. Construct the line through D parallel to AB. 4. Continue segment BC to meet that parallel line at E. BE is perpendicular to to DE by vertical angles, and angle DCE is equal to α because it is the complement of the complement of α. 5. Choose point F so that the segment AF is perpendicular to the line through D and E (from step 3). Angle ADF is equal to α+β by vertical angles.6. Fixing the length of AD at 1, the rest follows by basic trigonometry.