Let be a triangle and let be a point in its interior. Lines , , intersect sides , , at , , , respectively. Then
if and only if lies on
: at least one of the medians of triangle .
: at least one of the altitudes of triangle .
: at least one of the angle bisector of triangle .
: at least one of the perpendicular side bisector of triangle .
(Here denotes the area of triangle .)
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