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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
1999 Ukraine Team Selection Test
5
5
Part of
1999 Ukraine Team Selection Test
Problems
(1)
pairs of equal angles in a convex pentagon wanted
Source: Ukrainian TST 1999 p5
2/13/2020
A convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
with
D
C
=
D
E
DC = DE
D
C
=
D
E
and
∠
D
C
B
=
∠
D
E
A
=
9
0
o
\angle DCB = \angle DEA = 90^o
∠
D
CB
=
∠
D
E
A
=
9
0
o
is given. Let
F
F
F
be a point on the segment
A
B
AB
A
B
such that
A
F
:
B
F
=
A
E
:
B
C
AF : BF = AE : BC
A
F
:
BF
=
A
E
:
BC
. Prove that
∠
F
C
E
=
∠
A
D
E
\angle FCE = \angle ADE
∠
FCE
=
∠
A
D
E
and
∠
F
E
C
=
∠
B
D
C
\angle FEC = \angle BDC
∠
FEC
=
∠
B
D
C
.
pentagon
equal angles
geometry