Bisector0

Triangle bisectors

The main properties of the (inner) bisector lines of the angles of a triangle are:
[1] Each is the geometric locus of points equidistant from the sides of the angle it bisects.
[2] The three bisectors intersect at a point I, center of the incircle of the triangle.
[3] The trace of a bisector on the opposite side divides it at ratio equal to the ratio of the sides  of the angle it  bisects.
[4] The same is true for the trace on the opposite side of the external bisector: A*B/A*C=AB/AC and similarly for the
      other traces B* and C* (of the external bisectors). Hence on each side the two traces (internal/external) of the
      bisectors are harmonic conjugate (see Harmonic.html ) with respect to the vertices on this  side.
[5] The pedal ( Pedal.html ) triangle I_aI_bI_c of I (its projections on the sides) has angles (π-A)/2, (π-B)/2, (π-C)/2.
[6] The external bisectors of the angles cut the opposite sides at three collinear points A*, B*, C*. The line
      containingA*,B*,C* is the trilinear polar (see TrilinearPolar.html ) of I.

[1] A point I on the bisector defines by its projections I_b, I_c on the sides two equal right angled triangles AII_b and AII_c. Thus
      II_b=II_c.
[2] Assume I is the intersection point of the bisectors at B and C. Then II_c=II_a (bisector at B) and II_a=II_b (bisector at C). Hence
      II_c = II_b i.e. I is on the bisector of A.
[3-4] Let B' be the intersection point of the parallel BB' to the bisector AD. Triangle ABB' is isosceles and by the parallelity of BB'
         to AD: BD/DC = B'A/AC = AB/AC. The proof for the external bisector is analogous. The statement on the harmonicity is a
         consequence.
[5] AI_cII_b is a cyclic quadrangle hence angle(I_cII_b) = π-A etc..
[6] Given that A*B/A*C = AB/AC, B*C/B*A=BC/BA, C*A/C*B=CA/CB, apply Menealaus theorem ( Menelaus.html ) etc.. The
     harmonicity cited is a sufficient condition for A*, B*, C* in order to be on the trilinear polar of I (see reference below). The
     triangle formed by the intersection points of two external and one internal bisector is examined in Bisector1.html .
     With the bisectors are related the incircle and the excircles of the triangle. Some properties of them are studied in
     the file Incircle.html .

Triangle bisectors

See Also