MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1991 India National Olympiad
2
2
Part of
1991 India National Olympiad
Problems
(1)
Prove an area
Source: INMO 1991 Problem 2
10/3/2005
Given an acute-angled triangle
A
B
C
ABC
A
BC
, let points
A
′
,
B
′
,
C
′
A' , B' , C'
A
′
,
B
′
,
C
′
be located as follows:
A
′
A'
A
′
is the point where altitude from
A
A
A
on
B
C
BC
BC
meets the outwards-facing semicircle on
B
C
BC
BC
as diameter. Points
B
′
,
C
′
B', C'
B
′
,
C
′
are located similarly. Prove that
A
[
B
C
A
′
]
2
+
A
[
C
A
B
′
]
2
+
A
[
A
B
C
′
]
2
=
A
[
A
B
C
]
2
A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2
A
[
BC
A
′
]
2
+
A
[
C
A
B
′
]
2
+
A
[
A
B
C
′
]
2
=
A
[
A
BC
]
2
where
A
[
A
B
C
]
A[ABC]
A
[
A
BC
]
is the area of triangle
A
B
C
ABC
A
BC
.
geometry
trigonometry