How to find the Fermat Point using the construction of the tangent to ellipse? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:14:46Z http://mathoverflow.net/feeds/question/2305 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2305/how-to-find-the-fermat-point-using-the-construction-of-the-tangent-to-ellipse How to find the Fermat Point using the construction of the tangent to ellipse? Vasile MoČ™oi 2009-10-24T15:55:58Z 2012-02-02T13:47:49Z <p>Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point). I want a hint for solving this problem using the construction of the tangent to ellipse. (Hadamard, Lesson de Geometrie Elementaire, II, problem no. 745).</p> http://mathoverflow.net/questions/2305/how-to-find-the-fermat-point-using-the-construction-of-the-tangent-to-ellipse/2625#2625 Answer by Philipp Lampe for How to find the Fermat Point using the construction of the tangent to ellipse? Philipp Lampe 2009-10-26T15:27:55Z 2009-10-26T15:43:11Z <p>I have the vague idea that Hadamard is referring to the construction where you erect equilateral triangles BCA', CAB' and ABC' on the sides of the triangle, as described <a href="http://www.cut-the-knot.org/Generalization/fermat%5Fpoint.shtml" rel="nofollow">here</a>. The Fermat point is the intersection of the cevians AA', BB' and CC'. It can also be constructed using the various angles of 60 resp. 120 degrees. </p> <p>In the construction of the tangent from a point P to an ellipse with foci F and F' (in the book you cite), they consider an additional point f. The correspondence should be </p> <p>F &harr; C,</p> <p>F' &harr; A,</p> <p>P &harr; B,</p> <p>f &harr; P'.</p> <p>The general philosophy behind both, I think, is to convert a sum of segments into a single segment. </p>