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2020 Stanford Mathematics Tournament
8
8
Part of
2020 Stanford Mathematics Tournament
Problems
(1)
SMT 2020 Geometry #8
Source:
8/9/2023
Consider an acute angled triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with side lengths
7
7
7
,
8
8
8
, and
9
9
9
. Let
H
H
H
be the orthocenter of
A
B
C
ABC
A
BC
. Let
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
, and
Γ
C
\Gamma_C
Γ
C
be the circumcircles of
△
B
C
H
\vartriangle BCH
△
BC
H
,
△
C
A
H
\vartriangle CAH
△
C
A
H
, and
△
A
B
H
\vartriangle ABH
△
A
B
H
respectively. Find the area of the region
Γ
A
∪
Γ
B
∪
Γ
C
\Gamma_A \cup \Gamma_B \cup \Gamma_C
Γ
A
∪
Γ
B
∪
Γ
C
(the set of all points contained in at least one of
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
, and
Γ
C
\Gamma_C
Γ
C
).
geometry