Important lines in triangle - reverse problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:18:25Z http://mathoverflow.net/feeds/question/56914 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56914/important-lines-in-triangle-reverse-problem Important lines in triangle - reverse problem Beni Bogosel 2011-02-28T18:12:31Z 2011-02-28T18:12:31Z <p>It is known that if three numbers \$x,y,z\$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length \$x,y,z\$. Also, if \$x,y,z>0\$ (no condition imposed) there exists a triangle with angle bisectors of length \$x,y,z\$ (the proof of this is very beautiful and uses Brouwer fixed point theorem).</p> <p>I was wondering if there are some other results like this:</p> <ul> <li>if \$x,y,z>0\$ satisfy the family of conditions \$ { C_1,C_2,...,C_n }\$(possibly void) then there exists a triangle for which the lengths of some important lines (for eg. symmedians) are \$x,y,z\$.</li> </ul> <p>Do you know any such results?</p>