Lecture
Antibisector of an angle of a triangle (from Lat. anti, bi- «double» and sectio «cutting») — a specific ray originating at the vertex of the angle that divides the angle into two angles.
The antibisector of an interior angle is the locus of points inside the angle whose distances to the two sides of the angle are inversely proportional to the squares of those sides.
In a triangle, the antibisector of an angle can also mean the segment of the antibisector of that angle up to its intersection with the opposite side.
Like bisectors, antibisectors can be drawn not only to the interior angles but also to the exterior angles of a triangle. In doing so, the property of their mutual isotomy or isotomic conjugacy is preserved.
Antibisectors of a triangle were first introduced by Ócagne (D’Ocagne).
We assume the definitions of an isogonal cevian and an isometric cevian to be known; we recall that the antibisector, the antisymmedian, and the antialtitude are the isometrics of the bisector, the symmedian, and the altitude of a triangle.
The following relation of Steiner (1828) for isogonal cevians is also known
and 
We will now prove that an analogous relation exists for isometric cevians.
Proposition
Consider triangle ABC. Let and be two isometric cevians, then the following relation holds:
Proof
figure 1
From the law of sines applied to triangles ABA1, ACA1, it follows (see the figure above)
From relations (1) and (2) we keep
The law of sines applied in triangles 
leads to
From relations (4) and (5) we obtain:
Because 
and 
are isometric cevians, from relations (3) and (6) we obtain relation (*) from the statement of the proposition.
Applications
1. If AA1 is the bisector in triangle ABC and 
is its isometric, that is, the antibisector, then from (*) we obtain
Taking into account the law of sines in triangle ABC, we obtain
2. If 
is the symmedian and 
is the antisymmedian, from (*) we obtain
Indeed, being a symmedian, is isogonal to the median AM and
3. If is the altitude in triangle ABC, 
and is its isometric (antialtitude), relation (*) becomes.
Indeed
therefore
From (*) we get
or
therefore
4. If 
is the isogonal of the antibisector 
then
(Maurice D’Ocagne, 1883)
Proof
The Steiner relation for 
and 
is
But 
is the bisector and by the bisector theorem 
but 
and 
therefore
and we obtain the D’Ocagne relation
5. If in triangle ABC the cevian 
is isogonal to the symmedian 
then
Proof
Since AA1 is the symmedian, from Steiner's relation we derive that
The Steiner relation for 
gives us
Taking into account the preceding relation, we obtain
6. If 
is the isogonal of the antialtitude 
in triangle ABC, in which the altitude AA1 has 
, then
Proof
If AA1 is the altitude in triangle ABC 
, then
Because 
is the antimedian, we have 
and 
then 
Observation
The preceding results can be generalized for anticevians of rank k and their isogonals.
antisymmedian
antialtitude
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