MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2024 China National Olympiad
5
5
Part of
2024 China National Olympiad
Problems
(1)
Perfect Hexagon in China MO
Source: CMO 2024 P5
11/29/2023
In acute
△
A
B
C
\triangle {ABC}
△
A
BC
,
K
{K}
K
is on the extention of segment
B
C
BC
BC
.
P
,
Q
P, Q
P
,
Q
are two points such that
K
P
∥
A
B
,
B
K
=
B
P
KP \parallel AB, BK=BP
K
P
∥
A
B
,
B
K
=
BP
and
K
Q
∥
A
C
,
C
K
=
C
Q
KQ\parallel AC, CK=CQ
K
Q
∥
A
C
,
C
K
=
CQ
. The circumcircle of
△
K
P
Q
\triangle KPQ
△
K
PQ
intersects
A
K
AK
A
K
again at
T
{T}
T
. Prove that: (1)
∠
B
T
C
+
∠
A
P
B
=
∠
C
Q
A
\angle BTC+\angle APB=\angle CQA
∠
BTC
+
∠
A
PB
=
∠
CQ
A
. (2)
A
P
⋅
B
T
⋅
C
Q
=
A
Q
⋅
C
T
⋅
B
P
AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP
A
P
⋅
BT
⋅
CQ
=
A
Q
⋅
CT
⋅
BP
.Proposed by Yijie He and Yijuan Yao
geometry
circumcircle