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Random Geometry Problems from Ukrainian Contests
Kharkiv City MO Seniors - geometry
Kharkiv City MO Seniors - geometry
Part of
Random Geometry Problems from Ukrainian Contests
Subcontests
(18)
2021.11.4
1
Hide problems
PQ bisects I_CL, excircle related, CD=r_c 2021 Kharkiv City MO 11.4
In the triangle
A
B
C
ABC
A
BC
, the segment
C
L
CL
C
L
is the angle bisector. The
C
C
C
-exscribed circle with center at the point
I
c
I_c
I
c
touches the side of the
A
B
AB
A
B
at the point
D
D
D
and the extension of sides
C
A
CA
C
A
and
C
B
CB
CB
at points
P
P
P
and
Q
Q
Q
, respectively. It turned out that the length of the segment
C
D
CD
C
D
is equal to the radius of this exscribed circle. Prove that the line
P
Q
PQ
PQ
bisects the segment
I
C
L
I_CL
I
C
L
.
2021.10.5
1
Hide problems
P, N, L collinear if KM//AC, incircle and a circle 2021 Kharkiv City MO 10.5
The inscribed circle
Ω
\Omega
Ω
of triangle
A
B
C
ABC
A
BC
touches the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
K
K
K
and
L
L
L
, respectively. The line
B
L
BL
B
L
intersects the circle
Ω
\Omega
Ω
for the second time at the point
M
M
M
. The circle
ω
\omega
ω
passes through the point
M
M
M
and is tangent to the lines
A
B
AB
A
B
and
B
C
BC
BC
at the points
P
P
P
and
Q
Q
Q
, respectively. Let
N
N
N
be the second intersection point of circles
ω
\omega
ω
and
Ω
\Omega
Ω
, which is different from
M
M
M
. Prove that if
K
M
∥
A
C
KM \parallel AC
K
M
∥
A
C
then the points
P
,
N
P, N
P
,
N
and
L
L
L
lie on one line.
2019.11.5
1
Hide problems
BP bisects segment DE, projections related (Kharkiv City XI 2019 Ukr)
In the acute-angled triangle
A
B
C
ABC
A
BC
, let
C
D
,
A
E
CD, AE
C
D
,
A
E
be the altitudes. Points
F
F
F
and
G
G
G
are the projections of
A
A
A
and
C
C
C
on the line
D
E
DE
D
E
, respectively,
H
H
H
and
K
K
K
are the projections of
D
D
D
and
E
E
E
on the line
A
C
AC
A
C
, respectively. The lines
H
F
HF
H
F
and
K
G
KG
K
G
intersect at point
P
P
P
. Prove that line
B
P
BP
BP
bisects the segment
D
E
DE
D
E
.
2019.10.5
1
Hide problems
K is the orthocenter of II_aE, excenter (Kharkiv City X 2019 Ukr)
In triangle
A
B
C
ABC
A
BC
, point
I
I
I
is incenter ,
I
a
I_a
I
a
is the
A
A
A
-excenter. Let
K
K
K
be the intersection point of the
B
C
BC
BC
with the external bisector of the angle
B
A
C
BAC
B
A
C
, and
E
E
E
be the midpoint of the arc
B
A
C
BAC
B
A
C
of the circumcircle of triangle
A
B
C
ABC
A
BC
. Prove that
K
K
K
is the orthocenter of triangle
I
I
a
E
II_aE
I
I
a
E
.
2018.11.4
1
Hide problems
collinearity wanted, 3 circumcircles related (Kharkiv City XI 2018 - Ukr)
The line
ℓ
\ell
ℓ
parallel to the side
B
C
BC
BC
of the triangle
A
B
C
ABC
A
BC
, intersects its sides
A
B
,
A
C
AB,AC
A
B
,
A
C
at the points
D
,
E
D,E
D
,
E
, respectively. The circumscribed circle of triangle
A
B
C
ABC
A
BC
intersects line
ℓ
\ell
ℓ
at points
F
F
F
and
G
G
G
, such that points
F
,
D
,
E
,
G
F,D,E,G
F
,
D
,
E
,
G
lie on line
ℓ
\ell
ℓ
in this order. The circumscribed circles of the triangles
F
E
B
FEB
FEB
and
D
G
C
DGC
D
GC
intersect at points
P
P
P
and
Q
Q
Q
. Prove that points
A
,
P
A, P
A
,
P
and
Q
Q
Q
are collinear.
2018.10.4
1
Hide problems
CT bisects MP, AM=AN,BM=BK, circumcircle related (Kharkiv City X 2018- Ukr)
On the sides
A
B
,
A
C
,
B
C
AB, AC ,BC
A
B
,
A
C
,
BC
of the triangle
A
B
C
ABC
A
BC
, the points
M
,
N
,
K
M, N, K
M
,
N
,
K
are selected, respectively, such that
A
M
=
A
N
AM = AN
A
M
=
A
N
and
B
M
=
B
K
BM = BK
BM
=
B
K
. The circle circumscribed around the triangle
M
N
K
MNK
MN
K
intersects the segments
A
B
AB
A
B
and
B
C
BC
BC
for the second time at points
P
P
P
and
Q
Q
Q
, respectively. Lines
M
N
MN
MN
and
P
Q
PQ
PQ
intersect at point
T
T
T
. Prove that the line
C
T
CT
CT
bisects the segment
M
P
MP
MP
.
2017.11.5
1
Hide problems
AB,CD,XY concurrent or //, cyclic, circumcircle (Kharkiv City XI 2017- Ukr)
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in the circle
ω
\omega
ω
. Lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
E
E
E
. Points
M
M
M
and
N
N
N
are selected on segments
A
D
AD
A
D
and
B
C
BC
BC
, respectively, so that
A
M
:
M
D
=
B
N
:
N
C
AM: MD = BN: NC
A
M
:
M
D
=
BN
:
NC
. The circumscribed circle of the triangle
E
M
N
EMN
EMN
intersects the circle
ω
\omega
ω
at points
X
X
X
and
Y
Y
Y
. Prove that the lines
A
B
,
C
D
AB, CD
A
B
,
C
D
and
X
Y
XY
X
Y
intersect at the same point or are parallel.
2017.10.4
1
Hide problems
<DAB=90^o,<ADC = < BAM => < ADB = CAM (Kharkiv City X 2017- Ukr)
In the quadrangle
A
B
C
D
ABCD
A
BC
D
, the angle at the vertex
A
A
A
is right. Point
M
M
M
is the midpoint of the side
B
C
BC
BC
. It turned out that
∠
A
D
C
=
∠
B
A
M
\angle ADC = \angle BAM
∠
A
D
C
=
∠
B
A
M
. Prove that
∠
A
D
B
=
∠
C
A
M
\angle ADB = \angle CAM
∠
A
D
B
=
∠
C
A
M
.
2016.11.5
1
Hide problems
circumcenter lies on circle, BK = AC, CL = AB (Kharkiv City XI 2016 - Ukr)
The circle
ω
\omega
ω
passes through the vertices
B
B
B
and
C
C
C
of triangle
A
B
C
ABC
A
BC
and intersects its sides
A
C
,
A
B
AC,AB
A
C
,
A
B
at points
A
,
E
A,E
A
,
E
, respectively. On the ray
B
D
BD
B
D
, a point
K
K
K
such that
B
K
=
A
C
BK = AC
B
K
=
A
C
is chosen , and on the ray
C
E
CE
CE
, a point
L
L
L
such that
C
L
=
A
B
CL = AB
C
L
=
A
B
is chosen. Prove that the center
O
O
O
of the circumscribed circle of the triangle
A
K
L
AKL
A
K
L
lies on the circle
ω
\omega
ω
.
2016.10.3
1
Hide problems
collinearity, circumcenter and circumircle related (Kharkiv City X 2016 - Ukr)
Let
A
D
AD
A
D
be the bisector of an acute-angled triangle
A
B
C
ABC
A
BC
. The circle circumscribed around the triangle
A
B
D
ABD
A
B
D
intersects the straight line perpendicular to
A
D
AD
A
D
that passes through point
B
B
B
, at point
E
E
E
. Point
O
O
O
is the center of the circumscribed circle of triangle
A
B
C
ABC
A
BC
. Prove that the points
A
,
O
,
E
A, O, E
A
,
O
,
E
lie on the same line.
2015.11.3
1
Hide problems
midpoint wanted, right angles inside a rectagle (Kharkiv City XI 2015 - Ukr)
In the rectangle
A
B
C
D
ABCD
A
BC
D
, point
M
M
M
is the midpoint of the side
B
C
BC
BC
. The points
P
P
P
and
Q
Q
Q
lie on the diagonal
A
C
AC
A
C
such that
∠
D
P
C
=
∠
D
Q
M
=
9
0
o
\angle DPC = \angle DQM = 90^o
∠
D
PC
=
∠
D
QM
=
9
0
o
. Prove that
Q
Q
Q
is the midpoint of the segment
A
P
AP
A
P
.
2015.10.3
1
Hide problems
AC = PQ, intersecting circumcircles (Kharkiv City X 2015 - Ukr)
On side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
, point
M
M
M
is selected. A straight line passing through
M
M
M
intersects the segment
A
C
AC
A
C
at point
N
N
N
and the ray
C
B
CB
CB
at point
K
K
K
. The circumscribed circle of the triangle
A
M
N
AMN
A
MN
intersects
ω
\omega
ω
, the circumscribed circle of the triangle
A
B
C
ABC
A
BC
, at points
A
A
A
and
S
S
S
. Straight lines
S
M
SM
SM
and
S
K
SK
S
K
intersect with
ω
\omega
ω
for the second time at points
P
P
P
and
Q
Q
Q
, respectively. Prove that
A
C
=
P
Q
AC = PQ
A
C
=
PQ
.
2014.11.5
1
Hide problems
right angle wanted, collinear given (Kharkiv City XI 2014 - Ukr)
In the convex quadrilateral of the
A
B
C
D
ABCD
A
BC
D
, the diagonals of
A
C
AC
A
C
and
B
D
BD
B
D
are mutually perpendicular and intersect at point
E
E
E
. On the side of
A
D
AD
A
D
, a point
P
P
P
is chosen such that
P
E
=
E
C
PE = EC
PE
=
EC
. The circumscribed circle of the triangle
B
C
D
BCD
BC
D
intersects the segment
A
D
AD
A
D
at the point
Q
Q
Q
. The circle passing through point
A
A
A
and tangent to the line
E
P
EP
EP
at point
P
P
P
intersects the segment
A
C
AC
A
C
at point
R
R
R
. It turns out that points
B
,
Q
,
R
B, Q, R
B
,
Q
,
R
are collinear. Prove that
∠
B
C
D
=
9
0
o
\angle BCD = 90^o
∠
BC
D
=
9
0
o
.
2014.10.4
1
Hide problems
angle chasing inside a square (Kharkiv City X 2014 - Ukr)
Let
A
B
C
D
ABCD
A
BC
D
be a square. The points
N
N
N
and
P
P
P
are chosen on the sides
A
B
AB
A
B
and
A
D
AD
A
D
respectively, such that
N
P
=
N
C
NP=NC
NP
=
NC
. The point
Q
Q
Q
on the segment
A
N
AN
A
N
is such that that
∠
Q
P
N
=
∠
N
C
B
\angle QPN=\angle NCB
∠
QPN
=
∠
NCB
. Prove that
∠
B
C
Q
=
1
2
∠
A
Q
P
\angle BCQ=\frac{1}{2}\angle AQP
∠
BCQ
=
2
1
∠
A
QP
.
2013.11.4
1
Hide problems
<AA_ K=<BB_1M when B_1K//BC, A_1M//AC (Kharkiv City XI 2013 - Ukr)
In the triangle
A
B
C
ABC
A
BC
, the heights
A
A
1
AA_1
A
A
1
and
B
B
1
BB_1
B
B
1
are drawn. On the side
A
B
AB
A
B
, points
M
M
M
and
K
K
K
are chosen so that
B
1
K
∥
B
C
B_1K\parallel BC
B
1
K
∥
BC
and
A
1
M
∥
A
C
A_1 M\parallel AC
A
1
M
∥
A
C
. Prove that the angle
A
A
1
K
AA_1K
A
A
1
K
is equal to the angle
B
B
1
M
BB_1M
B
B
1
M
.
2013.10.4
1
Hide problems
tangent circles wanted, cyclic pentagon related (Kharkiv City X 2013 - Ukr)
The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in the circle
ω
\omega
ω
. Let
T
T
T
be the intersection point of the diagonals
B
E
BE
BE
and
A
D
AD
A
D
. A line is drawn through the point
T
T
T
parallel to
C
D
CD
C
D
, which intersects
A
B
AB
A
B
and
C
E
CE
CE
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that the circumscribed circle of the triangle
A
X
Y
AXY
A
X
Y
is tangent to
ω
\omega
ω
.
2012.10.4
1
Hide problems
midpoint is foot of altitude (Kharkiv City X 2012 - Ukr)
In the acute-angled triangle
A
B
C
ABC
A
BC
on the sides
A
C
AC
A
C
and
B
C
BC
BC
, points
D
D
D
and
E
E
E
are chosen such that points
A
,
B
,
E
A, B, E
A
,
B
,
E
, and
D
D
D
lie on one circle. The circumcircle of triangle
D
E
C
DEC
D
EC
intersects side
A
B
AB
A
B
at points
X
X
X
and
Y
Y
Y
. Prove that the midpoint of segment
X
Y
XY
X
Y
is the foot of the altitude of the triangle, drawn from point
C
C
C
.
2012.11.4
1
Hide problems
parallel wanted, incircle related (Kharkiv City XI 2012 - Ukr)
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touches its sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
at points
D
,
E
D, E
D
,
E
and
E
E
E
, respectively. Point
G
G
G
lies on circle
ω
\omega
ω
in such a way that
F
G
FG
FG
is a diameter. Lines
E
G
EG
EG
and
F
D
FD
F
D
intersect at point
H
H
H
. Prove that
A
B
∥
C
H
AB \parallel CH
A
B
∥
C
H
.