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Contests
National and Regional Contests
Ukraine Contests
Random Geometry Problems from Ukrainian Contests
Ukrainian Geometry Olympiad
2017 Ukrainian Geometry Olympiad
2017 Ukrainian Geometry Olympiad
Part of
Ukrainian Geometry Olympiad
Subcontests
(4)
4
3
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right angle wanted starting with a parallelogram and one circumcircle
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and
P
P
P
be an arbitrary point of the circumcircle of
Δ
A
B
D
\Delta ABD
Δ
A
B
D
, different from the vertices. Line
P
A
PA
P
A
intersects the line
C
D
CD
C
D
at point
Q
Q
Q
. Let
O
O
O
be the center of the circumcircle
Δ
P
C
Q
\Delta PCQ
Δ
PCQ
. Prove that
∠
A
D
O
=
9
0
o
\angle ADO = 90^o
∠
A
D
O
=
9
0
o
.
concyclic wanted, incircle, circumcircle, midline related in a right triangle
In the right triangle
A
B
C
ABC
A
BC
with hypotenuse
A
B
AB
A
B
, the incircle touches
B
C
BC
BC
and
A
C
AC
A
C
at points
A
1
{{A}_{1}}
A
1
and
B
1
{{B}_{1}}
B
1
respectively. The straight line containing the midline of
Δ
A
B
C
\Delta ABC
Δ
A
BC
, parallel to
A
B
AB
A
B
, intersects its circumcircle at points
P
P
P
and
T
T
T
. Prove that points
P
,
T
,
A
1
P,T,{{A}_{1}}
P
,
T
,
A
1
and
B
1
{{B}_{1}}
B
1
lie on one circle.
tangent circumcircles wanted, symmedian related
Let
A
D
AD
A
D
be the inner angle bisector of the triangle
A
B
C
ABC
A
BC
. The perpendicular on the side
B
C
BC
BC
at the point
D
D
D
intersects the outer bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
at point
I
I
I
. The circle with center
I
I
I
and radius
I
D
ID
I
D
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
F
F
F
and
E
E
E
respectively.
A
A
A
-symmedian of
Δ
A
F
E
\Delta AFE
Δ
A
FE
intersects the circumcircle of
Δ
A
F
E
\Delta AFE
Δ
A
FE
again at point
X
X
X
. Prove that the circumcircles of
Δ
A
F
E
\Delta AFE
Δ
A
FE
and
Δ
B
X
C
\Delta BXC
Δ
BXC
are tangent.
3
2
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fixed circle, starting with two intersecting circles and an orthocenter
Circles
w
1
,
w
2
{w}_{1},{w}_{2}
w
1
,
w
2
intersect at points
A
1
{{A}_{1}}
A
1
and
A
2
{{A}_{2}}
A
2
. Let
B
B
B
be an arbitrary point on the circle
w
1
{{w}_{1}}
w
1
, and line
B
A
2
B{{A}_{2}}
B
A
2
intersects circle
w
2
{{w}_{2}}
w
2
at point
C
C
C
. Let
H
H
H
be the orthocenter of
Δ
B
A
1
C
\Delta B{{A}_{1}}C
Δ
B
A
1
C
. Prove that for arbitrary choice of point
B
B
B
, the point
H
H
H
lies on a certain fixed circle.
AP is tangent to the circumcircle of BLP, starting with a right triangle
On the hypotenuse
A
B
AB
A
B
of a right triangle
A
B
C
ABC
A
BC
, we denote a point
K
K
K
such that
B
K
=
B
C
BK = BC
B
K
=
BC
. Let
P
P
P
be a point on the perpendicular from the point
K
K
K
to line
C
K
CK
C
K
, equidistant from the points
K
K
K
and
B
B
B
. Let
L
L
L
be the midpoint of
C
K
CK
C
K
. Prove that line
A
P
AP
A
P
is tangent to the circumcircle of
Δ
B
L
P
\Delta BLP
Δ
B
L
P
.
2
2
Hide problems
equal angles related to a trapezoid
Point
M
M
M
is the midpoint of the base
B
C
BC
BC
of trapezoid
A
B
C
D
ABCD
A
BC
D
. On base
A
D
AD
A
D
, point
P
P
P
is selected. Line
P
M
PM
PM
intersects line
D
C
DC
D
C
at point
Q
Q
Q
, and the perpendicular from
P
P
P
on the bases intersects line
B
Q
BQ
BQ
at point
K
K
K
. Prove that
∠
Q
B
C
=
∠
K
D
A
\angle QBC = \angle KDA
∠
QBC
=
∠
KD
A
.
angle chasing, double angle given, perpendicularity wanted
On the side
A
C
AC
A
C
of a triangle
A
B
C
ABC
A
BC
, let a
K
K
K
be a point such that
A
K
=
2
K
C
AK = 2KC
A
K
=
2
K
C
and
∠
A
B
K
=
2
∠
K
B
C
\angle ABK = 2 \angle KBC
∠
A
B
K
=
2∠
K
BC
. Let
F
F
F
be the midpoint of
A
C
AC
A
C
,
L
L
L
be the projection of
A
A
A
on
B
K
BK
B
K
. Prove that
F
L
⊥
B
C
FL \bot BC
F
L
⊥
BC
.
1
1
Hide problems
angle chasing with 2 midpoints, equal angles given and wanted
In the triangle
A
B
C
ABC
A
BC
,
A
1
{{A}_{1}}
A
1
and
C
1
{{C}_{1}}
C
1
are the midpoints of sides
B
C
BC
BC
and
A
B
AB
A
B
respectively. Point
P
P
P
lies inside the triangle. Let
∠
B
P
C
1
=
∠
P
C
A
\angle BP {{C}_{1}} = \angle PCA
∠
BP
C
1
=
∠
PC
A
. Prove that
∠
B
P
A
1
=
∠
P
A
C
\angle BP {{A}_{1}} = \angle PAC
∠
BP
A
1
=
∠
P
A
C
.