MathDB

2023.7

1

A,X,Y,E are concyclic iff AB=BC, given BC/AC=3/2

ABC

in which holds

|BC|: |AC| = 3:

2. Let

D

be the midpoint of the side

\overline{AC}

, and P the midpoint of the segment

\overline{BD}

. A point

X

is given on the line

AC

so that

|AX| = |BC|

, where

A

is between

X

and

C

. The line

XP

intersects the side

\overline{BC}

at point

E

. The line

DE

intersects the line

AP

at point

Y

. Prove that the points

A

,

X

,

Y

,

E

lie on one circle if and only if

|AB| = |BC|

.

2023.3

1

cyclic hexagon wanted with opposite sides // and not equal

A convex hexagon

ABCDEF

is given, with each two opposite sides of different lengths and parallel (

AB \parallel DE

,

BC \parallel EF

and

CD \parallel FA

). If

|AE| = |BD|

and

|BF| = |CE|

, prove that the hexagon

ABCDEF

is cyclic.

2022.7

1

TG _|_ T F , touchpoints of incircle fof isosceles related

In the triangle

ABC

holds

|AB| = |AC|

and the inscribed circle touches the sides

\overline{BC}

,

\overline{AC}

and

\overline{AB}

at the points

D

,

E

and

F

respectively . The perpendicular from the point

D

to the line

EF

intersects the side

\overline{AB}

at the point

G

, and the circles circumscribed around the triangles

AEF

and

ABC

intersect at the points

A

and

T

. Prove that the lines

T G

and

T F

are perpendicular.

2022.3

1

A lies on angle bisector of <QLM

Let

ABC

be an acute-angled triangle in which

|AB| < |AC|

and let circle

k

with center

O

be its circumscribed circle. Let

P

and

Q

be points on sides

\overline{BC}

and

\overline{AB}

respectively such that

AQPO

is a parallelogram. Let

K

and

L

be the intersections of the perpendicular bisector of

\overline{OP}

with circle

k

, where

K

is on the shorter arc

AB

. Let

M

be the second intersection of the line

KQ

with the circle

k

. Prove that the point

A

belongs to the bisector of the angle

\angle QLM

.

2020.7

1

perpendicular wanted, AR=RQ, <CAP =<ABC, circle of diameter

A circle of diameter

AB

is given. There are points

C

and

D

on this circle, on different sides of the diameter such that holds

AC <BC

or

AC<AD

. The point

P

lies on the segment

BC

and

\angle CAP = \angle ABC

. The perpendicular from the point

C

to the line

AB

intersects the direction

BD

at the point

Q

. Lines

PQ

and

AD

intersect at point

R

, and the lines

PQ

and

CD

intersect at point

T

. If

AR=RQ

, prove that the lines

AT

and

PQ

are perpendicular.

2020.3

1

right angle wanted, incircle, circumcircle, perpendicular diagonals

Given a triangle

ABC

such that

AB<AC

. On sides

AB

and

BC

, points

P

and

Q

are marked respectively such that the lines

AQ

and

CP

are perpendicular, and the circle inscribed in the triangle

ABC

touches the length

PQ

. The line

CP

intersects the circle circumscribed around the triangle

ABC

at the points

C

and

T

. If the lines

CA,PQ

and

BT

intersect at one point, prove that the angle

\angle CAB

is right.

2010.7

1

AP passes through circumcenter of ABC, points symmetric wrt sides

Given a non- isosceles triangle

ABC

. Let the points

B'

and

C'

be symmetric to the points

B

and

C

wrt

AC

and

AB

respectively. If the circles circumscribed around triangles

ABB'

and

ACC'

intersect at point

P

, prove that the line

AP

passes through the center of the circumcircle of the triangle

ABC

.

2010.3

1

<BAP =<CAQ wanted, <BAE = <CAF , EP//CD , FQ //CD

Let

D

be a point on the side

AC

of triangle

ABC

. Let

E

and

F

be points on the segments

BD

and

BC

respectively, such that

\angle BAE = \angle CAF

. Let

P

and

Q

be points on the segments

BC

and

BD

respectively, such that

EP \parallel CD

and

FQ \parallel CD

. Prove that

\angle BAP = \angle CAQ

.

2012.7

1

MNPQ is cyclic iff ABC is right triangle (touchpoints of incircle,2 excircles)

Let the points

M

and

N

be the intersections of the inscribed circle of the right-angled triangle

ABC

, with sides

AB

and

CA

respectively , and points

P

and

Q

respectively be the intersections of the ex-scribed circles opposite to vertices

B

and

C

with direction

BC

. Prove that the quadrilateral

MNPQ

is a cyclic if and only if the triangle

ABC

is right-angled with a right angle at the vertex

A

.

2012.3

1

AN x NC = CD x BN wanted, inside cyclic ABCD with AD=BD

Let

ABCD

be a cyclic quadrilateral such that

|AD| =|BD|

and let

M

be the intersection of its diagonals. Furthermore, let

N

be the second intersection of the diagonal

AC

with the circle passing through points

B, M

and the center of the circle inscribed in triangle

BCM

. Prove that

AN \cdot NC = CD \cdot BN

2013.7

1

angle of feet of angle bisectors wanted if <B=120^o

In triangle

ABC

, the angle at vertex

B

is

120^o

. Let

A_1, B_1, C_1

be points on the sides

BC, CA, AB

respectively such that

AA_1, BB_1, CC_1

are bisectors of the angles of triangle

ABC

. Determine the angle

\angle A_1B_1C_1

.

2013.3

1

AFA_1C is cyclic iff BF bisects CG, orthocenter, parallelogram related

Given a pointed triangle

ABC

with orthocenter

H

. Let

D

be the point such that the quadrilateral

AHCD

is parallelogram. Let

p

be the perpendicular to the direction

AB

through the midpoint

A_1

of the side

BC

. Denote the intersection of the lines

p

and

AB

with

E

, and the midpoint of the length

A_1E

with

F

. The point where the parallel to the line

BD

through point

A

intersects

p

denote by

G

. Prove that the quadrilateral

AFA_1C

is cyclic if and only if the lines

BF

passes through the midpoint of the length

CG

.

2014.7

1

tangent lines at given youchpoints wanted, incenter, circumcircle related

Let point

I

be the center of the inscribed circle of an acute-angled triangle

ABC

. Rays

AI

and

BI

intersect the circumcircle

k

of triangle

ABC

at points

D

and

E

respectively. The segments

DE

and

CA

intersect at point

F

, the line through point

E

parallel to the line

FI

intersects the circle

k

at point

G

, and the lines

FI

and

DG

intersect at point

H

. Prove that the lines

CA

and

BH

touch the circumcircle of the triangle

DFH

at the points

F

and

H

respectively.

2014.3

1

angle bisector <BDC$ // AS, angle bisector of <BAC, midpoints, // related

Given a triangle

ABC

in which

|AB|>|AC|

. Let

P

be the midpoint of the side

BC

, and

S

the point in which the angle bisector of

\angle BAC

intersects that side. The parallel with the line

AS

through the point

P

intersects lines

AB

and

AC

at points

X

and

Y

respectively . Let

Z

be the point be such that

Y

is the midpoint of the length

XZ

and let the lines

BY

and

CZ

intersect at point

D

. Prove that the angle bisector of

\angle BDC

is parallel to the lines

AS

.

2015.7

1

D lies on the circumcircle of XYZ wanted, 2 circumcircles, altitudes

In an acute-angled triangle

ABC

is

AB > BC

, and the points

A_1

and

C_1

are the feet of the altitudes of from the vertices

A

and

C

. Let

D

be the second intersection of the circumcircles of triangles

ABC

and

A_1BC_1

(different of

B

). Let

Z

be the intersection of the tangents to the circumcircle of the triangle ABC at the points

A

and

C

, and let the lines

ZA

and

A_1C_1

intersect at the point

X

, and the lines

ZC

and

A_1C_1

intersect at the point

Y

. Prove that the point

D

lies on the circumcircle of the triangle

XYZ

.

2015.3

1

concurrency wanted, line secant to both intersecting circles related

Circles

k_1

and

k_2

intersect at points

M

and

N

. The line

\ell

intersects the circle

k_1

at points

A

and

C

, the circle

K_2

at points

B

and

D

so that the points

A,B,C

and

D

lie on the line

\ell

are in that order. Let

X

a point on the line

MN

such that the point

M

is located between the points

X

and

N

. Let

P

be the intersection of lines

AX

and

BM

, and

Q

be the intersection of lines

DX

and

CM

. If

K

is the midpoint of segment

AD

and

L

is the midpoint of segment

BC

, prove that the lines

XK

and

ML

intersect on the line

PQ

.

2016.3

1

3 feet of altitudes and another point concyclic wanted, cyclic given, AC//DE//BF

Given a cyclic quadrilateral

ABCD

such that the tangents at points

B

and

D

to its circumcircle

k

intersect at the line

AC

. The points

E

and

F

lie on the circle

k

so that the lines

AC, DE

and

BF

parallel. Let

M

be the intersection of the lines

BE

and

DF

. If

P, Q

and

R

are the feet of the altitides of the triangle

ABC

, prove that the points

P, Q, R

and

M

lie on the same circle

2017.7

1

AD/ MD + BE/ ME + CF/ MF >= 9/2 , M interior, circumcircles related

The point

M

is located inside the triangle

ABC

. The ray

AM

intersects the circumcircle of the triangle

MBC

once more at point

D

, the ray

BM

intersects the circumcircle of the triangle

MCA

once more at point

E

, and the ray

CM

intersects the circumcircle of the triangle

MAB

once more at point

F

. Prove that holds

\frac{AD}{MD}+\frac{BE}{ME} +\frac{CF}{MF}\ge \frac92

2017.3

1

<IMA=<INB wanted, incenter, midpoint and arc midpoint related

In triangle

ABC

,

|AB| <|BC|

holds. Point

I

is the center of the circle inscribed in that triangle. Let

M

be the midpoint of the side

AC

, and

N

be the midpoint of the arc

AC

of the circumcircle of that triangle containing point

B

. Prove that

\angle IMA = \angle INB

.

2018.7

1

AI=IE wanted, incenter, circumcircle, arc midpoint, symm.of incenter wrt side

Given an acute-angled triangle

ABC

in which

|AB| <|AC|

. Point

D

is the midpoint of the shorter arc

BC

of its circumcircle. The point

I

is the center of its incircle, and the point

J

is symmetric point of

I

wrt line

BC

. The line

DJ

intersects the circumcircle of the triangle

ABC

at the point

E

belonging to the arc

AB

. Prove that

|AI |= |IE|

.

2019.7

1

min AB: CD wanted, diagonals AC, BD bisect DX, CX respectively in cyclic ABCD

On the side

AB

of the cyclic quadrilateral

ABCD

there is a point

X

such that diagonal

AC

bisects the segment

DX

, and the diagonal

BD

bisects the segment

CX

. What is the smallest possible ratio

|AB | : |CD|

in such a quadrilateral ?

2019.3

1

concyclic wanted, <PXM=<PYM, PA// BC, starting with isosceles

Given an isosceles triangle

ABC

such that

|AB|=|AC|

. Let

M

be the midpoint of the segment

BC

and let

P

be a point other than

A

such that

PA\parallel BC

. The points

X

and

Y

are located respectively on rays

PB

and

PC

, so that the point

B

is between

P

and

X

, the point

C

is between

P

and

Y

and

\angle PXM=\angle PYM

. Prove that the points

A,P,X

and

Y

are concyclic.

2018.3

1

Circumcenter is a reflection of another circumcenter

Let

k

be a circle centered at

O

. Let

\overline{AB}

be a chord of that circle and

M

its midpoint. Tangent on

k

at points

A

and

B

intersect at

T

. The line

\ell

goes through

T

, intersect the shorter arc

AB

at the point

C

and the longer arc

AB

at the point

D

, so that

|BC| = |BM|

. Prove that the circumcenter of the triangle

ADM

is the reflection of

O

across the line

AD

2016.7

1

common incircle of two triangles

Let

P

be a point inside a triangle

ABC

such that

\frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .

Lines

AP

,

BP

,

CP

intersect the circumcircle of triangle

ABC

again in

A'

,

B'

,

C'

. Prove that the triangles

ABC

and

A'B'C'

have a common incircle.

2011.7

1

Croatian mathematical olympiad, day 2 problem 3

Let

K

and

L

be the points on the semicircle with diameter

AB

. Denote intersection of

AK

and

AL

as

T

and let

N

be the point such that

N

is on segment

AB

and line

TN

is perpendicular to

AB

. If

U

is the intersection of perpendicular bisector of

AB

an

KL

and

V

is a point on

KL

such that angles

UAV

and

UBV

are equal. Prove that

NV

is perpendicular to

KL

.

2011.3

1

Croatian mathematical olympiad, day 1 problem 3

Triangle

ABC

is given with its centroid

G

and cicumcentre

O

is such that

GO

is perpendicular to

AG

. Let

A'

be the second intersection of

AG

with circumcircle of triangle

ABC

. Let

D

be the intersection of lines

CA'

and

AB

and

E

the intersection of lines

BA'

and

AC

. Prove that the circumcentre of triangle

ADE

is on the circumcircle of triangle

ABC

.

Croatia MO (HMO) - geometry

Subcontests

A,X,Y,E are concyclic iff AB=BC, given BC/AC=3/2

cyclic hexagon wanted with opposite sides // and not equal

TG _|_ T F , touchpoints of incircle fof isosceles related

A lies on angle bisector of <QLM

perpendicular wanted, AR=RQ, <CAP =<ABC, circle of diameter

right angle wanted, incircle, circumcircle, perpendicular diagonals

AP passes through circumcenter of ABC, points symmetric wrt sides

<BAP =<CAQ wanted, <BAE = <CAF , EP//CD , FQ //CD

MNPQ is cyclic iff ABC is right triangle (touchpoints of incircle,2 excircles)

AN x NC = CD x BN wanted, inside cyclic ABCD with AD=BD

angle of feet of angle bisectors wanted if <B=120^o

AFA_1C is cyclic iff BF bisects CG, orthocenter, parallelogram related

tangent lines at given youchpoints wanted, incenter, circumcircle related

angle bisector <BDC$ // AS, angle bisector of <BAC, midpoints, // related

D lies on the circumcircle of XYZ wanted, 2 circumcircles, altitudes

concurrency wanted, line secant to both intersecting circles related

3 feet of altitudes and another point concyclic wanted, cyclic given, AC//DE//BF

AD/ MD + BE/ ME + CF/ MF >= 9/2 , M interior, circumcircles related

<IMA=<INB wanted, incenter, midpoint and arc midpoint related

AI=IE wanted, incenter, circumcircle, arc midpoint, symm.of incenter wrt side

min AB: CD wanted, diagonals AC, BD bisect DX, CX respectively in cyclic ABCD

concyclic wanted, <PXM=<PYM, PA// BC, starting with isosceles

Circumcenter is a reflection of another circumcenter

common incircle of two triangles

Croatian mathematical olympiad, day 2 problem 3

Croatian mathematical olympiad, day 1 problem 3

Croatia MO (HMO) - geometry

Subcontests

A,X,Y,E are concyclic iff AB=BC, given BC/AC=3/2

cyclic hexagon wanted with opposite sides // and not equal

TG _|_ T F , touchpoints of incircle fof isosceles related

A lies on angle bisector of &lt;QLM

perpendicular wanted, AR=RQ, &lt;CAP =&lt;ABC, circle of diameter

right angle wanted, incircle, circumcircle, perpendicular diagonals

AP passes through circumcenter of ABC, points symmetric wrt sides

&lt;BAP =&lt;CAQ wanted, &lt;BAE = &lt;CAF , EP//CD , FQ //CD

MNPQ is cyclic iff ABC is right triangle (touchpoints of incircle,2 excircles)

AN x NC = CD x BN wanted, inside cyclic ABCD with AD=BD

angle of feet of angle bisectors wanted if &lt;B=120^o

AFA_1C is cyclic iff BF bisects CG, orthocenter, parallelogram related

tangent lines at given youchpoints wanted, incenter, circumcircle related

angle bisector &lt;BDC$ // AS, angle bisector of &lt;BAC, midpoints, // related

D lies on the circumcircle of XYZ wanted, 2 circumcircles, altitudes

concurrency wanted, line secant to both intersecting circles related

3 feet of altitudes and another point concyclic wanted, cyclic given, AC//DE//BF

AD/ MD + BE/ ME + CF/ MF &gt;= 9/2 , M interior, circumcircles related

&lt;IMA=&lt;INB wanted, incenter, midpoint and arc midpoint related

AI=IE wanted, incenter, circumcircle, arc midpoint, symm.of incenter wrt side

min AB: CD wanted, diagonals AC, BD bisect DX, CX respectively in cyclic ABCD

concyclic wanted, &lt;PXM=&lt;PYM, PA// BC, starting with isosceles

Circumcenter is a reflection of another circumcenter

common incircle of two triangles

Croatian mathematical olympiad, day 2 problem 3

Croatian mathematical olympiad, day 1 problem 3

A lies on angle bisector of <QLM

perpendicular wanted, AR=RQ, <CAP =<ABC, circle of diameter

<BAP =<CAQ wanted, <BAE = <CAF , EP//CD , FQ //CD

angle of feet of angle bisectors wanted if <B=120^o

angle bisector <BDC$ // AS, angle bisector of <BAC, midpoints, // related

AD/ MD + BE/ ME + CF/ MF >= 9/2 , M interior, circumcircles related

<IMA=<INB wanted, incenter, midpoint and arc midpoint related

concyclic wanted, <PXM=<PYM, PA// BC, starting with isosceles