Let be the incenter of a triangle and , , and be the intersections of the segments , , with the incircle. Also let the centroid lie inside the incircle and , , and be the intersections of the segments , , with the incircle. Then the perimeter of is less than or equal to that of , as proposed by Garfunkel (1981, 1982) and proved by Nyugen and Dergiades (2004).
Garfunkel's Inequality
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References
Garfunkel, J. "Problem 648." Crux Math. 7, 178, 1981.Garfunkel, J. "Solution to Problem 648." Crux Math. 8, 180-182, 1982.Nyugen, M. H. and Dergiades, N. "Garfunkel's Inequality." Forum Geom. 4, 153-156, 2004. http://forumgeom.fau.edu/FG2004volume4/FG200419index.html.Referenced on Wolfram|Alpha
Garfunkel's InequalityCite this as:
Weisstein, Eric W. "Garfunkel's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GarfunkelsInequality.html