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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2007 China Second Round Olympiad
1
1
Part of
2007 China Second Round Olympiad
Problems
(1)
a condition for 4 points to be concyclic
Source: China second round 2007 p1
3/10/2012
In an acute triangle
A
B
C
ABC
A
BC
,
A
B
<
A
C
AB<AC
A
B
<
A
C
.
A
D
AD
A
D
is the altitude dropped onto
B
C
BC
BC
and
P
P
P
is a point on
A
D
AD
A
D
. Let
P
E
⊥
A
C
PE\perp AC
PE
⊥
A
C
at
E
E
E
,
P
F
⊥
A
B
PF\perp AB
PF
⊥
A
B
at
F
F
F
and let
J
,
K
J,K
J
,
K
be the circumcentres of triangles
B
D
F
,
C
D
E
BDF, CDE
B
D
F
,
C
D
E
respectively. Prove that
J
,
K
,
E
,
F
J,K,E,F
J
,
K
,
E
,
F
are concyclic if and only if
P
P
P
is the orthocentre of triangle
A
B
C
ABC
A
BC
.
geometry
power of a point