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National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2019 Belarus Team Selection Test
7.1
7.1
Part of
2019 Belarus Team Selection Test
Problems
(1)
Just angles
Source: 2019 Belarus Team Selection Test 7.1
9/2/2019
The internal bisectors of angles
∠
D
A
B
\angle DAB
∠
D
A
B
and
∠
B
C
D
\angle BCD
∠
BC
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at the point
X
1
X_1
X
1
, and the external bisectors of these angles intersect at the point
X
2
X_2
X
2
. The internal bisectors of angles
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
C
D
A
\angle CDA
∠
C
D
A
intersect at the point
Y
1
Y_1
Y
1
, and the external bisectors of these angles intersect at the point
Y
2
Y_2
Y
2
. Prove that the angle between the lines
X
1
X
2
X_1X_2
X
1
X
2
and
Y
1
Y
2
Y_1Y_2
Y
1
Y
2
equals the angle between the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
.(A. Voidelevich)
geometry