[alogo] 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.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]
[3_0] [3_1] [3_2] [3_3]


[HonsEpis] Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995, p. 1.

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