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National and Regional Contests
Russia Contests
239 Open Math Olympiad
2018 239 Open Mathematical Olympiad
10-11.5
10-11.5
Part of
2018 239 Open Mathematical Olympiad
Problems
(1)
239 Open M0, 2018, Senior League, Problem 5
Source: 239 Open M0, 2018, Senior League, Problem 5
4/4/2023
Given a trapezoid
A
B
C
D
ABCD
A
BC
D
, with
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. Lines
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
E
E
E
, and lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
F
F
F
. It turns out that the circle with diameter
E
F
EF
EF
is tangent to the midline of the trapezoid. Prove that there exists a square such that there is a mutual correspondence between all six lines containing pairs of its vertices, and points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
, and
F
F
F
: each line corresponds to a point lying on it. Proposed by V. Mokin
geometry