MathDB
Problems
Contests
International Contests
Kvant Problems
Kvant 2022
M2717
M2717
Part of
Kvant 2022
Problems
(1)
Nice collinearity
Source: Kvant Magazine No. 9 2022 M2717
3/8/2023
In an acute triangle
A
B
C
ABC
A
BC
the heights
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
intersecting at
H
H{}
H
. Let
O
O{}
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
. The tangents to the circle
(
A
B
C
)
(ABC)
(
A
BC
)
drawn at
B
B{}
B
and
C
C{}
C
intersect at
T
T{}
T
. Let
K
K{}
K
and
L
L{}
L
be symmetric to
O
O{}
O
with respect to
A
B
AB
A
B
and
A
C
AC
A
C
respectively. The circles
(
D
F
K
)
(DFK)
(
D
F
K
)
and
(
D
E
L
)
(DEL)
(
D
E
L
)
intersect at a point
P
P{}
P
different from
D
D{}
D
. Prove that
P
,
D
P, D
P
,
D
and
T
T{}
T
lie on the same line.Proposed by Don Luu (Vietnam)
Kvant
geometry