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National and Regional Contests
China Contests
(China) National High School Mathematics League
2021 China Second Round A2
1
1
Part of
2021 China Second Round A2
Problems
(1)
AT bisects BC
Source: 2021 China Second Round A2 , additional test, p1
4/29/2024
As shown in the figure, in the acute angle
△
A
B
C
\vartriangle ABC
△
A
BC
,
A
B
>
A
C
AB > AC
A
B
>
A
C
,
M
M
M
is the midpoint of the minor arc
B
C
BC
BC
of the circumcircle
Ω
\Omega
Ω
of
△
A
B
C
\vartriangle ABC
△
A
BC
.
K
K
K
is the intersection point of the bisector of the exterior angle
∠
B
A
C
\angle BAC
∠
B
A
C
and the extension line of
B
C
BC
BC
. From point
A
A
A
draw a line perpendicular on
B
C
BC
BC
and take a point
D
D
D
(different from
A
A
A
) on that line , such that
D
M
=
A
M
DM = AM
D
M
=
A
M
. Let the circumscribed circle of
△
A
D
K
\vartriangle ADK
△
A
DK
intersect the circle
Ω
\Omega
Ω
at point
A
A
A
and at another point
T
T
T
. Prove that
A
T
AT
A
T
bisects line segment
B
C
BC
BC
. https://cdn.artofproblemsolving.com/attachments/1/3/6fde30405101620828d63ae31b8c0ffcec972f.png
geometry
bisects segment