Bisector

Bisector properties

Some further properties from those examined in Bisector1.html :
[1] The bisector AI intersects the circumcircle k of triangle ABC at F, the middle of arc BC.
[2] The middles of the sides of the triangle of excenters I_aI_bI_c coincide with the intersection points of these sides with the circumcircle.
[3] |FB|=|FC|=|FI| and similar equalities for the other points analogous to F.
[4] angle(AFM) = angle(B)-angle(C).

[1] That F is the middle of arc BC follows from the equality of angles ABF=FAC. Besides quadrilateral IBEC is cyclic, since IBE and ICE are right angles.
[2] Since I_aI_bI_c has ABC as its orthic. Thus k being the Euler circle of I_aI_bI_c.
[3] Follows from the previous two statements.
[4] Consider the symmetric B' of B w.r. to the bisector AI: angle(AFM)=angle(B'BC)=B-C.

With the bisectors of a triangle are connected the following two important theorems:
a) Theorem of Feuerbach: That the Euler circle of a triangle is tangent to its inner and its three excircles.
b) Theorem of Steiner-Lehmus: That equality of two inner bisectors implies that the triangle is isosceles.
With the bisectors are connected also some difficult (or impossible) to resolve triangle constructions. An interesting case is examined in the file TriangleBisectorConstruction.html .
Besides, another important related subject that adds much structure into the geometry of the triangle is the one on the isogonal conjugation.

Bisector properties

See Also