BL = 2.76 cm
LC = 7.61 cm
AB = 8.05 cm
AC = 8.43 cm
m∠BAL = 19.01°
m∠LAC = 58.93°
A
AB·sin(m∠BAL)
= 0.36
AC·sin(m∠LAC)
BL
= 0.36
LC
h
B
L
C
Lemma: If vertex A of triangle ABC is joined to any point L on
BC, then BL/LC = (ABsin(BAL) / ACsin(LAC)).
AC = 2.84 cm
Cross Ratios
AC
CD
= 0.35
AB
BD
CD = 4.75 cm
AB = 4.81 cm
BD = 2.78 cm
HG = 1.54 cm
GE = 3.00 cm
HF = 2.72 cm
HG
GE
= 0.35
HF
FE
FE = 1.82 cm
A
H
D
B
C
G
E
F
V
Theorem: The cross ratio of any pencil of four distinct lines is
equal to the cross ratio of the corresponding four points in
which any ordinary transversal cuts the pencil.
V
D
A
C
B
AC = 4.85 cm
CB = 2.38 cm
AD = 14.23 cm
BD = 7.00 cm
DB = –7.00 cm
AC
= 2.03
CB
AD
= –2.03
DB
AC
CB
= –1.00
AD
DB
C
A
D
B
Definition: A system of points on a straight line is called a range.
Given a range of points A, B, C, and D on a straight line, we say
that segment AB is divided harmonically if (AC/CB)/(AD/DB) = -1.
A, B, C, and D are said to form an harmonic range. That is, if a
segment is divided internally and externally in the same ratio, it is
said to be divided harmonically. The aforementioned ratio is
usually written
{ AB, CD}.
t
l
E
m
AC = 4.94 cm
CB = 1.37 cm
AD = 8.75 cm
BD = 2.43 cm
G
C
A
D
B
F BD = 1.22 cm
BG = 1.22 cm
AC
= 3.60
CB
AD
= 3.60
BD
C
A
O
D
B
If AB is divided harmonically at C and D, and if O is
the midpoint of AB, then (OC)(OD) = OB^2 , and
conversely.
A
B
O
Divide AB harmonically at C and D.
r = 2.26 cm
OC = 1.27 cm
C
A
B
O
r2
= 4.00 cm
OC
r = 2.43 cm
r2 = 5.93 cm2
OC = 1.48 cm
AC = 3.92 cm
CB = 0.95 cm
AD = 6.43 cm
DB = 1.56 cm
C
A
O
B
D
AC
CB
= 1.00
AD
DB
OD = 4.00 cm
Inversion Script