MathDB

Regional Olympiad - FBH 2015 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018

Let

ABC

be a triangle with incenter

I

. Line

AI

intersects circumcircle of

ABC

in points

A

and

D

,

(A \neq D)

. Incircle of

ABC

touches side

BC

in point

E

. Line

DE

intersects circumcircle of

ABC

in points

D

and

F

,

(D \neq F)

. Prove that

\angle AFI = 90^{\circ}

geometryincentercircumcircle

Regional Olympiad - FBH 2015 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018

In parallelogram

ABCD

holds

AB=BD

. Let

K

be a point on

AB

, different from

A

, such that

KD=AD

. Let

M

be a point symmetric to

C

with respect to

K

, and

N

be a point symmetric to point

B

with respect to

A

. Prove that

DM=DN

geometryparallelogram

Regional Olympiad - FBH 2015 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018

Let

F

be an intersection point of altitude

CD

and internal angle bisector

AE

of right angled triangle

ABC

,

\angle ACB = 90^{\circ}

. Let

G

be an intersection point of lines

ED

and

BF

. Prove that area of quadrilateral

CEFG

is equal to area of triangle

BDG

geometryangle bisector

Regional Olympiad - FBH 2015 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018

Let

O

and

I

be circumcenter and incenter of triangle

ABC

. Let incircle of

ABC

touches sides

BC

,

CA

and

AB

in points

D

,

E

and

F

, respectively. Lines

FD

and

CA

intersect in point

P

, and lines

DE

and

AB

intersect in point

Q

. Furthermore, let

M

and

N

be midpoints of

PE

and

QF

. Prove that

OI \perp MN

geometrycircumcircleincenter

3

Problems(4)