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Turkey Team Selection Test
2019 Turkey Team SeIection Test
7
7
Part of
2019 Turkey Team SeIection Test
Problems
(1)
TST Geometry P7
Source: Turkey Team Selection Test 2019 Day 3 Problem 7
3/26/2019
In a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
9
0
∘
\angle ACB = 90^{\circ}
∠
A
CB
=
9
0
∘
D
D
D
is the foot of the altitude of
C
C
C
. Let
E
E
E
and
F
F
F
be the reflections of
D
D
D
with respect to
A
C
AC
A
C
and
B
C
BC
BC
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of
△
E
C
B
\triangle {ECB}
△
ECB
and
△
F
C
A
\triangle {FCA}
△
FC
A
. Show that:
2
O
1
O
2
=
A
B
2O_1O_2=AB
2
O
1
O
2
=
A
B
geometry