(thing) by IWhoSawTheFace Wed Nov 16 2011 at 18:58:21

The angle bisector theorem states that when an angle bisector is drawn from the vertex of a triangle, it splits the opposite side into lengths that are proportionate to the lengths of the sides adjacent to the vertex. This was first shown by Euclid in his Elements. It's not something I was taught in high school geometry, but it's enormously useful. If you're an engineer who uses geometry infrequently, you may find this theorem useful as well. 


                    C
                   /|\
                  /  |   \
                 /    |     \
                /      |       \      b
            a  /        |         \    
              /          |           \
             /            |             \
            /              |               \
           /                |                 \
          /                  |                   \ 
         B--------------------D--------------------A
                  x                    y

Angle Bisector Theorem: If ∠BCD = ∠ACD, then a/x = b/y.

Another useful form is that ratio of the the lengths of the two portions of the side split by D is proportionate to the lengths of the adjacent sides: x/y = a/b.

Euclid showed this in Book VI, Proposition 3.

This theorem is key to understanding Ceva's Theorem, and to concepts like the point of concurrency which are cornerstones to modern Euclidean geometry.

The "generalized" angle bisector theorem is for the case where the vertex angle is bisected into unequal angles. In that case x/y = (a*sin∠BCD)/(b*sin∠ACD).

Everything2 Writeups: Articles on (topic)

  1. IWhoSawTheFace, Stewart's Theorem, Nov, 2011
  2. IWhoSawTheFace, Triangle and Circle Geometry, (Nov, 2011 - not finished yet)
  3. IWhoSawTheFace, Cevian, Nov, 2011

References: Useful books and references on geometry

  1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
    § 1.4, “The Medians and the Centroid,” p. 10
    § 1.5, “The Incircle and the Circumcircle,” pp. 11-16
    § 1.6, “The Euler Line and the Orthocenter,” p. 17
    § 1.7, “The Nine Point Circle,” pp. 18-20
    § 1.9, “Morley’s Theorem,” pp. 23-25
    § 1.6, “The Euler Line and the Orthocenter,” p. 17
  2. Dan Pedoe, Geometry: A Comprehensive Course
  3. C. Stanley Ogilvy, Excursions in Geometry, (c) 1969
    An elegant, thin discourse on geometry.
  4. J.L. Heilbron, Geometry Civilized, ©2000
  5. David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
  6. Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
    Ch. 4, Geometry,
    § 4.5.1, “Triangles,” p. 271
    § 4.6, “Circles,” p. 277

Internet References

  1. Angle Bisection on GSP. Gives you a glimpse of Geometer's Sketchpad and how it's used to give insight into Euclidean theorems.
  2. Wikipedia, "Angle Bisector Theorem"
  3. D. Joyce, Euclid's Elements, Book 4, Proposition 3, "To circumscribe a circle about a given triangle." David Joyce is a professor of Mathematics and Computer Science at Clark University. He rendered Euclid's Elements into HTML, added Java applets to illustrate geometric constructions with live, movable points and lines. If you're a geometry buff, you should bookmark this site.
  4. Weisstein, Eric W. "Angle Bisector" From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W. "Exterior Angle Bisector" From MathWorld--A Wolfram Web Resource. This will be useful when discussing excircles.
  6. Jim Loy, "Centers of Triangles," Univ. of Georgia. I've bookmarked this page and have almost committed its contents to memory. It's a nice one-page summary of the rudiments of triangle geometry.
  7. Alexander Bogomolny, "Angle Bisector Theorem,” From Cut The Knot--mathematical topics.

I like it!