MathDB
Problems
Contests
International Contests
IberoAmerican
2022 Iberoamerican
5
5
Part of
2022 Iberoamerican
Problems
(1)
Interesting geo with tangents, external bisector and many equal segments
Source: Iberoamerican 2022, Day 2, P2
9/29/2022
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcircle
Γ
\Gamma
Γ
. Let
P
P
P
and
Q
Q
Q
be points in the half plane defined by
B
C
BC
BC
containing
A
A
A
, such that
B
P
BP
BP
and
C
Q
CQ
CQ
are tangents to
Γ
\Gamma
Γ
and
P
B
=
B
C
=
C
Q
PB = BC = CQ
PB
=
BC
=
CQ
. Let
K
K
K
and
L
L
L
be points on the external bisector of the angle
∠
C
A
B
\angle CAB
∠
C
A
B
, such that
B
K
=
B
A
,
C
L
=
C
A
BK = BA, CL = CA
B
K
=
B
A
,
C
L
=
C
A
. Let
M
M
M
be the intersection point of the lines
P
K
PK
P
K
and
Q
L
QL
Q
L
. Prove that
M
K
=
M
L
MK=ML
M
K
=
M
L
.
geometry