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National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P8
P8
Part of
2023 China Team Selection Test
Problems
(1)
Finding out 3 circles
Source: 2023 China TST Problem 8
3/18/2023
In non-isosceles acute
△
A
B
C
{}{\triangle ABC}
△
A
BC
,
A
P
AP
A
P
,
B
Q
BQ
BQ
,
C
R
CR
CR
is the height of the triangle.
A
1
A_1
A
1
is the midpoint of
B
C
BC
BC
,
A
A
1
AA_1
A
A
1
intersects
Q
R
QR
QR
at
K
K
K
,
Q
R
QR
QR
intersects a straight line that crosses
A
{A}
A
and is parallel to
B
C
BC
BC
at point
D
{D}
D
, the line connecting the midpoint of
A
H
AH
A
H
and
K
{K}
K
intersects
D
A
1
DA_1
D
A
1
at
A
2
A_2
A
2
. Similarly define
B
2
B_2
B
2
,
C
2
C_2
C
2
.
△
A
2
B
2
C
2
{}\triangle A_2B_2C_2
△
A
2
B
2
C
2
is known to be non-degenerate, and its circumscribed circle is
ω
\omega
ω
. Prove that: there are circles
⊙
A
′
\odot A'
⊙
A
′
,
⊙
B
′
\odot B'
⊙
B
′
,
⊙
C
′
\odot C'
⊙
C
′
tangent to and INSIDE
ω
\omega
ω
satisfying: (1)
⊙
A
′
\odot A'
⊙
A
′
is tangent to
A
B
AB
A
B
and
A
C
AC
A
C
,
⊙
B
′
\odot B'
⊙
B
′
is tangent to
B
C
BC
BC
and
B
A
BA
B
A
, and
⊙
C
′
\odot C'
⊙
C
′
is tangent to
C
A
CA
C
A
and
C
B
CB
CB
. (2)
A
′
A'
A
′
,
B
′
B'
B
′
,
C
C
C
' are different and collinear. Created by Sihui Zhang
geometry
China TST