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National and Regional Contests
Canada Contests
Canada National Olympiad
1970 Canada National Olympiad
2
2
Part of
1970 Canada National Olympiad
Problems
(1)
Obtuse triangle
Source: Canada 1970, Problem 2
5/14/2006
Given a triangle
A
B
C
ABC
A
BC
with angle
A
A
A
obtuse and with altitudes of length
h
h
h
and
k
k
k
as shown in the diagram, prove that
a
+
h
≥
b
+
k
a+h\ge b+k
a
+
h
≥
b
+
k
. Find under what conditions
a
+
h
=
b
+
k
a+h=b+k
a
+
h
=
b
+
k
. [asy] size(6cm);pair A = dir(105), C = dir(170), B = dir(10), D = foot(B, A, C), E = foot(A, B, C);draw(A--B--C--cycle); draw(B--D--A--E);dot(A); dot(B); dot(C); dot(D); dot(E);label("
A
A
A
", A, dir(110)); label("
B
B
B
", B, B); label("
C
C
C
", C, C); label("
D
D
D
", D, D); label("
E
E
E
", E, dir(45));label("
h
h
h
", A--E, dir(0)); label("
k
k
k
", B--D, dir(45)); [/asy]
trigonometry