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Menelaus's theorem
If a line meets the sides
B
C
,
C
A
BC, CA
B
C
,
C
A
and
A
B
AB
A
B
of a triangle in points
P
,
Q
P, Q
P
,
Q
and
R
R
R
then the product of the ratios
B
P
P
C
.
C
Q
Q
A
.
A
R
R
B
\Large \frac {BP}{PC}.\frac {CQ}{QA}.\frac {AR}{RB}
P
C
B
P
.
Q
A
C
Q
.
R
B
A
R
is -
1
Conversely, if points
P
,
Q
P, Q
P
,
Q
and
R
R
R
are on the three sides of the triangle and this ratio is -
1
then the three points are collinear.