KöMaL 2018 C1463

$M$ is an interior point of a regular triangle $ABC$. The feet of the perpendiculars dropped from $M$ onto the sides $AB$, $BC$ and $CA$ are $H$, $K$ and $P$, respectively. Prove that $(i)$ ${|AH|}^2+{|BK|}^2+{|CP|}^2= {|HB|}^2+{|KC|}^2+{|PA|}^2$; $(ii)$ $|AH|+|BK|+|CP|= |HB|+|KC|+|PA|$.