Bisector1

The triangle formed by the bisectors

By an argument similar to one defining the intersection point I of the three internal bisectors (see Bisector0.html ),
two external bisectors and one internal intersect at the same point. Such a point is equidistant from the sides of thetriangle. Thus there exists three excircles with centers at these intersection points I_a, I_b, I_c.
[1] The triangle I_aI_bI_c formed by the external bisectors of ABC has the internal bisectors as altitudes.
[2] The original triangle is the orthic of I_aI_bI_c. Its circumcircle is the Euler cirlce of I_aI_bI_c.
[3] Let J_a, J_a, J_b be the projections on the sides of the incenter I. Let further J_a_a, J_a_b, J_a_c be the   projections of the excenter J_a on the sides and define similarly J_b_a, J_b_b, etc.. The following relations hold:
  [3.1]  |AJ_a_c|=|AJ_a_b|=s, where s=(a+b+c)/2 the half-perimeter of the triangle.
  [3.2]  |CJ_a_b|=|CJ_a_a|=s-b, |AJ_c|=|AJ_b|=s-a.
  [3.3]  |BJ_a|=|CJ_a_a|=s-b, |J_aJ_a_a|=b-c.
  [3.4]  BC and J_aJ_a_a have the same middle.
[4]  r^-1 = r_a^-1+r_b^-1+r_c^-1.
      Here a, b, c denote the length of BC, CA, AB respectively. Similar formulas hold by replacing the symbols with      a cyclic permutation of a,b,c. Here the symbols {r, r_a, r_b, r_c} denote the radii of the in/excircles respectively.

[1] The orthogonality of inner to outer bisector at C, B, A is trivial.
[2] Trivial consequence of [1] (see Orthic.html ).
[3] All relations are easy consequences of the fact that the tangents to a circle from a point are equal.
[4] Measure the area of ABC in terms of the in/exradii: area(ABC) = s*r. Taking the triangles with bases the sides of ABC and one vertex at I_b: 2*area(ABC) = (c+a-b)*r_b. Hence r/r_b = (c+a-b)/(2*s). To prove the property add the three corresponding formulas, resulting by cyclic permutation of a, b, c.
[5] The formulas proved here are used in Fundamental_Invariants.html to derive expressions like f.e. the sum of squares a² + b² + c² in terms of s, r and R (which are precisely the fundamental invariants of the triangle ABC).

Some further remarks related to the bisectors are contained in Bisector.html .

The triangle formed by the bisectors

See Also