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National and Regional Contests
France Contests
French Mathematical Olympiad
1990 French Mathematical Olympiad
Problem 5
Problem 5
Part of
1990 French Mathematical Olympiad
Problems
(1)
maximum area involving A-excircle
Source: France 1990 P5
5/18/2021
In a triangle
A
B
C
ABC
A
BC
,
Γ
\Gamma
Γ
denotes the excircle corresponding to
A
A
A
,
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
are the points of tangency of
Γ
\Gamma
Γ
with
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively, and
S
(
A
B
C
)
S(ABC)
S
(
A
BC
)
denotes the region of the plane determined by segments
A
B
′
,
A
C
′
AB',AC'
A
B
′
,
A
C
′
and the arc
C
′
A
′
B
′
C'A'B'
C
′
A
′
B
′
of
Γ
\Gamma
Γ
.Prove that there is a triangle
A
B
C
ABC
A
BC
of a given perimeter
p
p
p
for which the area of
S
(
A
B
C
)
S(ABC)
S
(
A
BC
)
is maximal. For this triangle, give an approximate measure of the angle at
A
A
A
.
geometry
Triangles