Ratios of lengths and ratios of areas are invariant under affine transformations, so the answer should hold for arbitrary triangles, not just equilateral ones. -- Gene From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, May 29, 2015 9:10 AM Subject: [math-fun] Triangle puzzle For s in [0,1], define an s-median of a triangle (ABC in counterclockwise order) from vertex A to be the line segment connecting A to the point P of the opposite side BC such that BP : PC = s : (1-s) . (E.g., an ordinary median is a (1/2)-median.) A / \ / \ / \ / \ / \ B------P---------C Old math puzzle: Suppose we draw all three s-medians of an equilateral triangle ABC, where s = 1/3. Let delta be the triangle bounded by segments of the three s-medians. Find the ratio area(delta) / area(ABC). New math puzzle: Suppose we draw the r-, s- and t-medians of an equilateral triangle ABC for r, s, t in [0,1], again assuming ABC go counterclockwise around the perimeter. Let delta be the triangle bounded by segments of the three "medians". Find the ratio area(delta) / area(ABC) . ——Dan