MathDB
Problems
Contests
International Contests
KoMaL A Problems
KoMaL A Problems 2018/2019
A. 754
A. 754
Part of
KoMaL A Problems 2018/2019
Problems
(1)
Nice Geometry
Source: KöMaL A. 754
3/19/2022
Let
P
P
P
be a point inside the acute triangle
A
B
C
,
ABC,
A
BC
,
and let
Q
Q
Q
be the isogonal conjugate of
P
.
P.
P
.
Let
L
,
M
L,M
L
,
M
and
N
N
N
be the midpoints of the shorter arcs
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
of the circumcircle of
A
B
C
,
ABC,
A
BC
,
respectively. Let
X
A
X_A
X
A
be the intersection of ray
L
Q
LQ
L
Q
and circle
(
P
B
C
)
,
(PBC),
(
PBC
)
,
let
X
B
X_B
X
B
be the intersection of ray
M
Q
MQ
MQ
and circle
P
C
A
,
PCA,
PC
A
,
and let
X
C
X_C
X
C
be the intersection of ray
N
Q
NQ
NQ
and circle
(
P
A
B
)
.
(PAB).
(
P
A
B
)
.
Prove that
P
,
X
A
,
X
B
P,X_A,X_B
P
,
X
A
,
X
B
and
X
C
X_C
X
C
are concyclic or coincide.Proposed by Gustavo Cruz (São Paulo)
geometry
komal