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Czech-Polish-Slovak Match
2018 Czech-Polish-Slovak Match
2
2
Part of
2018 Czech-Polish-Slovak Match
Problems
(1)
Circumcircles of ABC, ADE, AO_1O_2 have a common point
Source: Czech-Polish-Slovak Match 2018, Problem 2
7/2/2018
Let
A
B
C
ABC
A
BC
be an acute scalene triangle. Let
D
D
D
and
E
E
E
be points on the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
D
=
C
E
BD=CE
B
D
=
CE
. Denote by
O
1
O_1
O
1
and
O
2
O_2
O
2
the circumcentres of the triangles
A
B
E
ABE
A
BE
and
A
C
D
ACD
A
C
D
, respectively. Prove that the circumcircles of the triangles
A
B
C
,
A
D
E
ABC, ADE
A
BC
,
A
D
E
, and
A
O
1
O
2
AO_1O_2
A
O
1
O
2
have a common point different from
A
A
A
.Proposed by Patrik Bak, Slovakia
geometry
circumcircle