Stewart's Theorem

A theorem in planar Euclidean geometry that gives an algebraic relationship between sides of a triangle and segments of sides when cevians divide a triangle in half. The theorem was discovered by the Scottish mathematician and minister, Matthew Stewart (1717/19 - 1785).

A cevian is a line segment whose one endpoint is the vertex of a triangle and the other endpoint lies somewhere along the opposite side. The length of the cevian can be computed from the knowledge of the lengths of the sides of the triangle, as well as the lengths of the subsegments of the side divided by the endpoint of the cevian.

Within triangle ΔABC construct a cevian connecting vertex A to point D on the side BC. Point D cuts side BC into segments CD and BD, having lengths x and y, respectively. Then if d is the length of AD, the following relationship holds:

b²y + c²x = a(d²+xy)

or

d = sqrt(((b²y+c²x)/a)-xy)

Medians: If the cevian is a median, it bisects side BC into equal length segments, and x = y. Then the cevian length is:

d = sqrt((x/a)*(b²y+c²x))

Angle Bisectors: If the cevian is an angle bisector, the by the angle bisector theorem, (y/x) = (a/b), or y = (a/b)*x. Then the cevian length can be found by

(b + c)² = a²*(1 + d²/(x*y))

Altitudes: If the cevian is an altitude, then it makes a right angle with side BC. Then apply the Pythagorean formula to find d as a function of side lengths and x and y:

d² = b² - x² = c² - y²

Everything2 Writeups: Articles on (topic)

1. eipi10, Theorem of Menelaus, Aug., 2002
2. eipi10, Ceva's Theorem, Aug, 2002
3. IWhoSawTheFace, Triangle and Circle Geometry, (Nov, 2011 - not finished yet)
4. IWhoSawTheFace, Giovanni Ceva, Nov, 2011
5. IWhoSawTheFace, Cevian, Nov, 2011
6. IWhoSawTheFace, Giovanna Ceva, Nov, 2011
7. IWhoSawTheFace, Menelaus, 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. Wikipedia, "Stewart's Theorem"
2. Weisstein, Eric W. "Stewart's Theorem" From MathWorld--A Wolfram Web Resource.
3. Alexander Bogomolny, "(Corollaries from) Pythagoras' Theorem" From Cut The Knot--mathematical topics. This has a section on Stewart's Theorem.