PEM Level C
6 February 2010
Topics in Geometry: Menelaus’ Theorem
Theorems:
1. (Menelaus’ Theorem) Let
ABC
be a triangle, and let
X
,
Y
, and
Z
be the feet of some cevians
from
A
,
B
, and
C
, respectively. If
X
,
Y
, and
Z
are collinear, then
AZ
ZB
·
BX
XC
·
CY
Y A
= 1
.
2. (Converse of Menelaus’ Theorem) Let
ABC
be a triangle, and let
X
,
Y
, and
Z
be the feet of
some cevians from
A
,
B
, and
C
, respectively. If
AZ
ZB
·
BX
XC
·
CY
Y A
= 1
,
then
X
,
Y
, and
Z
are collinear.
Exercises:
1. Triangle
ABC
has
∠
B
= 90
◦
,
BC
= 3, and
AB
= 4. Point
D
is on segment
AC
such that
AD
= 1, and point
E
is the midpoint of
AB
. Join
D
and
E
by a segment, and extend
DE
to meet the extended
CB
at
F
. Find
BF
.
2. In
4
ABC
, let
X
and
Y
be points on segments
BC
and
AC
, respectively, and
R
the point
of intersection of
AX
and
BY
. Given that
AY/Y C
=
p
and
AR/RX
=
q
, where 0
< p < q
,
express
BX
:
XC
in terms of
p
and
q
.
3. The diagonals
AC
and
BD
of the convex quadrilateral
ABCD
intersect at point
M
in such
a way that
AM
=
MC
and
DM
= 2
MB
. Suppose that
X
and
Y
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 Spring '09
 B
 Geometry, Pythagorean Theorem, Trigraph, triangle, AZ BX CY, ZB XC

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