Another solution to P6: As $EM$ is perpendicular on $AC$ we can say by perpendicular lemma, $CE^2 + AM^2 = AE^2 + CM^2$ or, $CM^2 - AM^2 = CE^2 - AE^2$ Again, $MF$ is perpendicular on $BC$ so, $CF^2 + BM^2 = CM^2 + BF^2$ or, $CM^2 - BM^2 = CF^2 - BF^2 = CM^2 - AM^2$ so, $CE^2 - AE^2 = CF^2 - BF^2$ ...