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National and Regional Contests
Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2015 Canadian Mathematical Olympiad Qualification
6
6
Part of
2015 Canadian Mathematical Olympiad Qualification
Problems
(1)
Right triangle, feet of altitude, angle bisector and median
Source: Canada Repêchage 2015/6
6/18/2016
Let
△
A
B
C
\triangle ABC
△
A
BC
be a right-angled triangle with
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
, and
A
B
<
A
C
AB < AC
A
B
<
A
C
. Let points
D
,
E
,
F
D, E, F
D
,
E
,
F
be located on side
B
C
BC
BC
such that
A
D
AD
A
D
is the altitude,
A
E
AE
A
E
is the internal angle bisector, and
A
F
AF
A
F
is the median.Prove that
3
A
D
+
A
F
>
4
A
E
3AD + AF > 4AE
3
A
D
+
A
F
>
4
A
E
.
geometry
angle bisector