Slanted Viviani, PWW
What Might This Be About?
$ABC$ is an equilateral triangle. Cevians $AD,$ $BE,$ $CF$ are equal (and, for the esthetics' sake, equally inclined to the corresponding sides.) Points $M,N,P$ are on $BC,AC,AB,$ respectively, such that $OM\parallel AD,$ $ON\parallel BE,$ and $OP\parallel CF.$
Prove that $OM+ON+OP=AD.$
The proof is supposed to be self-explanatory.
In case of difficulties in interpretation, please, have a look at a more explicit variant.
The illustration is by Grégoire Nicollier.