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MATH 348: Solved Emergency Problems, Week 4
Leonid Chindelevitch
August 26, 2008
1
Monday, July 28, 2008
1.
Problem:
From each of the centers
O
1
,O
2
of two circles
ω
1
,ω
2
tangents
are drawn to the other circle. Prove that equal chords are intercepted on
the circumferences.
Solution:
Let
O
1
E,O
1
F
be the tangents to
ω
2
, let
O
2
G,O
2
H
be the
tangents to
ω
1
, and let
A,B
and
C,D
be the intersections of these tangents
with
ω
1
and
ω
2
, respectively. Let
M
and
N
be the midpoints of
AB
and
CD
, respectively.
The line
O
1
O
2
is the bisector of the angles
AO
1
B
and
CO
2
D
, so that
M
and
N
lie on it. The rightangled triangles
O
1
EO
2
and
O
1
MA
are similar
by two angles, and so are
O
1
GO
2
and
CNO
2
. We deduce that
EO
2
O
1
O
2
=
MA
O
1
A
;
O
1
G
O
1
O
2
=
CN
CO
2
.
From these equalities, we now get
AB
= 2
AM
= 2
O
1
A
·
O
2
E
O
1
O
2
= 2
O
1
G
·
O
2
C
O
1
O
2
= 2
CN
=
CD.
It follows that the two chords,
AB
and
CD
, are indeed equal.
1
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View Full Document2.
Problem:
Prove that if perpendiculars are drawn from the feet of the
altitudes of a triangle to each of the other two sides, the six feet of these
perpendiculars lie on a circle.
Solution:
This circle is known as the
Taylor circle
of a triangle. For a
solution, see the National Mathematics Magazine, Vol. 18, No. 1 (Oct.
1943), p.40, or email me.
3.
Problem:
Prove that if, in a triangle, two medians are perpendicular,
then the three medians are the sides of a rightangled triangle.
Solution:
Let Δ
ABC
be the given triangle, let
D,E,F
be the midpoints
of the sides
a,b,c
respectively, and let
G
be the centroid. Let us deﬁne
x
:=
GD,y
:=
GE
; since the centroid divides the medians in a 1:2 ratio,
we have
GA
= 2
x,GB
= 2
y
. Assume that the medians
AD
and
BE
are
perpendicular; by Pythagoras’ theorem in the triangles
AGB,AGE,BGD
,
we obtain:
(2
x
)
2
+ (2
y
)
2
=
c
2
;
x
2
+ (2
y
)
2
= (
a/
2)
2
; (2
x
)
2
+
y
2
= (
b/
2)
2
.
From the formula for the length of the median proven in practice problem
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 Summer '06
 Karigiannis
 Geometry, Chords

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