Angle Bisectors and Triangles

The internal and external bisectors of a triangle have several interesting properties. We will mention a few of them here. You can find a brief statement about the origin of bisect at this link.
The first figure shows the three internal bisectors of the angles at A, B, and C and make clear the first important property.  The internal angle bisectors of any triangle intersect in a single point, and this point is the center of the inscribed circle.  This means that the perpendicular distance from the incenter to the sides of the triangle is the same distance, the radius of the incenter.  The radius of the inscribed triangle is interrelated with the area, A,  and the perimeter, P, of the triangle by the following equation.  A=PR/2.  Since one half the perimeter is often called the semi-perimeter, S, we can also write this as A=SR.

If we extend two sides of the triangle we can bisect the exterior angles in the same way.  In the figure the exterior angle at A is bisected by the ray AA'.  The External bisectors at A and C and the internal bisector at B all intersect in a common point also, and this point is the center of a circle that is tangent to the three sides also.  This is called an Excircle, and the center is the excenter.  Since there are three sides, this could be done three ways, so we label them by the bisector they are on.  The radius of this circle is also related to the area and perimeter of the triangle.  If we call the radius of this excircle Rb, then the A=(S-b)Rb. (remember S is the semi-perimeter, and b is the side adjacent to this excircle)   This could be repeated for any of the three excircles.

If the construction is repeated for all four circles we see that there are really three internal bisectors, and three external bisectors as the two external bisectors at a vertex lie on the same line.  The six lines intersect in sets of three in four distinct points, the incenter, and the three excenters.   The internal angle bisectors are perpendicular to the external bisectors through the same vertex.  This makes each of the internal angle bisectors an altitude of the triangle formed by the three excenters.  The point I then, which is the incenter of triangle ABC, is also the orthocenter of triangle Ea, Eb, Ec.    Another interesting fact about this figure is that the three excircles are all externally tangent to the  Nine Point Circle (not shown in the figure), and the incircle is internally tangent to it.

The radii of the four circles shown are related by a single equation, , and they also are related to the area of triangle ABC by the formula A=. If we ler r stand for the radius of the circle that just circumscibes triangle ABC, we get one more interesting theorem, .

If we mark the point where a bisector intersects the opposite side and measure the lengths, we discover another property of angle bisectors. In the figure the internal bisector from A crosses side a at the point A'1.  The external bisector crosses side a (extended) at A'2.  The ratio of BA' to A'C is equal to the ratio of c/b.  Notice that I did not put a subscript on the A'.  None is needed because the relationship is the same whether you put A'1 or A'2 in the relation.  (Where would A'2 be if the triangle were isosceles?)

As long as we are measuring, what can we find about the length of the angle bisector from A to A'1. We have labeled this "t" on the figure, and the two parts that make up side a are labeled x and y. We will use these in our explanation.  The length of the bisecting segment, t, squared  is equal to the product of b and c minus the product of x and y.  As an equation we write t2 = bc - xy.  This is a corollary of  Stewart's Thm.

Here is one additional property of the angle bisector that I have never seen in a textbook.  Each bisector, such as AA'1 is divided into two pieces by the incenter, I.  The lengths of these two pieces are always in a ratio related to the three sides.  Using the figure as a guide for our example, the ratio AI : IA'1 = (b+c) : a .