Cleaver

Cleaver line

This is defined to be a line from the middle of one side of the triangle t = (ABC), F middle of BC say, to a point I dividing the perimeter in two equal parts: here |FB|+|BI| = |IA|+|AC|. The following is true:
(1) FI is parallel to the bisector AH of angle A.
(2) The line IG, orthogonal to AB at the "cleaver point" I, passes through the middle G of the arc BAC of the circumcircle of triangle t.
(3) The three cleaver-lines, corresponding to the middles of the three different sides intersect at a point K.
(4) K is the incenter of the medial triangle s, whose vertices are the middles of the sides of t.