MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2018 Latvia Baltic Way TST
P9
P9
Part of
2018 Latvia Baltic Way TST
Problems
(1)
Triangle configuration with line intersection on circumcircle
Source: 2018 Latvia BW TST P9
3/26/2022
Acute triangle
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
<
A
C
AB<AC
A
B
<
A
C
, circumcircle
Γ
\Gamma
Γ
and circumcenter
O
O
O
is given. Midpoint of side
A
B
AB
A
B
is
D
D
D
. Point
E
E
E
is chosen on side
A
C
AC
A
C
so that
B
E
=
C
E
BE=CE
BE
=
CE
. Circumcircle of triangle
B
D
E
BDE
B
D
E
intersects
Γ
\Gamma
Γ
at point
F
F
F
(different from point
B
B
B
). Point
K
K
K
is chosen on line
A
O
AO
A
O
satisfying
B
K
⊥
A
O
BK \perp AO
B
K
⊥
A
O
(points
A
A
A
and
K
K
K
lie in different half-planes with respect to line
B
E
BE
BE
). Prove that the intersection of lines
D
F
DF
D
F
and
C
K
CK
C
K
lies on
Γ
\Gamma
Γ
.
geometry
circumcircle