Aim: To understand Stewart’s
Theorem
DO NOW: TAKE OUT CALCULATOR
DEFINE CEVIAN
A cevian is a straight line that
passes through a vertex of a
triangle or tetrahedron and
intersects the opposite side or
face.
Stewart’s Theorem
 Matthew Stewart (1717–1785) published this
theorem as Proposition II on page 2 in his 1746 book.
 It is said that Archimedes used the theorem though
(300 B.C.)
 In geometry, Stewart's theorem yields a relation
between a lengths of the sides of the triangle and the
length of a cevian of the triangle.
b2m+c2n=a(d2+mn)
b
c
d
n
m
a
2
2
2
b m + c n = a(d +m*n )
A=25
B=34
C=13
D=17
M=20
N=5
B
13
34
A
5
17
D
20
25
C
man+dad=bmb+cnc
b2m+c2n=amn+d2a
This is also often written , a form which invites
mnemonic memorization, e.g. "A man and his
dad put a bomb in the sink."
man+dad=bmb+cnc
D
D = 15 cm
A = 16 cm
B = 17 cm
C = 17 cm
M = 8 cm
N = 8 cm
Penny, Quinn, and Rachel all go to the same school. Penny
lives 5.5399 miles from Quinn, who lives 5.0803 miles from
Rachel. If the school is between Penny and Rachel and Penny
is 5.7073 miles away from it and Rachel is 2.2927 miles away
from it in the opposite direction, how far is Quinn from the
school?
Quinn
5.5399
x
Penny
5.7073
5.0803
2.2927
School
Rachel
Quinn
5.5399
3.75
x
Penny
5.7073
5.0803
2.2927
School
b2m+c2n=a(d2+mn)
Rachel
A = 18
B = 17
C = 16
M=8
N = 10
17
16
D = 13.8
8
10
If the cevian happens to be a
median, angle bisector, or altitude
the following formulas can be used
MEDIAN
ANGLE BISECTOR
ALTITUDE
Sources
1. HTTP://EASYCALCULATION.COM/THEORE
2.
3.
4.
5.
MS/STEWART-OR-APOLLONIUS.PHP
HTTP://MATH.STACKEXCHANGE.COM/QUE
STIONS/46616/PROVING-STEWARTSTHEOREM-WITHOUT-TRIG
HTTP://WWW.ANSWERS.COM/TOPIC/CEVI
AN-MATHEMATICS
HTTP://WWW.ARTOFPROBLEMSOLVING.C
OM/WIKI/INDEX.PHP/STEWART'S_THEOR
EM
HTTP://WWW.MS.UKY.EDU/~DROYSTER/C
OURSES/FALL11/MA341/CLASSNOTES/LEC
TURE%2013%20HANDOUTS.PDF