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National and Regional Contests
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Official Ukraine Selection Cycle
Kyiv City MO
Kyiv City MO - geometry
Kyiv City MO Juniors 2003+ geometry
2016.9.5
2016.9.5
Part of
Kyiv City MO Juniors 2003+ geometry
Problems
(1)
AC = AA_1 if <ABC = 45^o, AA_1= _|_ CC_1 (2016 Kyiv City MO 9.5)
Source:
9/5/2020
On the sides
B
C
BC
BC
and
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
the points
A
1
{{A} _ {1}}
A
1
and
C
1
{{C} _ {1}}
C
1
are selected accordingly so that the segments
A
A
1
A {{A} _ {1}}
A
A
1
and
C
C
1
C {{C} _ {1}}
C
C
1
are equal and perpendicular. Prove that if
∠
A
B
C
=
45
∘
\angle ABC = 45 {} ^ \circ
∠
A
BC
=
45
∘
, then
A
C
=
A
A
1
AC = A {{A} _ {1}}
A
C
=
A
A
1
.(Gogolev Andrew)
geometry
equal segments
perpendicularity