The midpoint (center) of the incircle is I. R is the midpoint of the red segment and G is the midpoint of the green segment. Show the RIG makes a line.
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Solution
Given triangle ABC and its incircle: draw B’Y’C’ parallel to BC so the incircle is now an excircle for AB’C’ with Y’ as contact point, so B’X’ = Y’C’. Draw line AY’Y and by similarity BX = YC. Now both R and G are on the parallel line to AY’Y through I and collinearity is proved (using the Triangle proportionality theorem in triangle XYA).