MathDB

2020 IGO Elementary P4

Source: 7th Iranian Geometry Olympiad (Elementary) P4

11/4/2020

Let

P

be an arbitrary point in the interior of triangle

\triangle ABC

. Lines

\overline{BP}

and

\overline{CP}

intersect

\overline{AC}

and

\overline{AB}

at

E

and

F

, respectively. Let

K

and

L

be the midpoints of the segments

BF

and

CE

, respectively. Let the lines through

L

and

K

parallel to

\overline{CF}

and

\overline{BE}

intersect

\overline{BC}

at

S

and

T

, respectively; moreover, denote by

M

and

N

the reflection of

S

and

T

over the points

L

and

K

, respectively. Prove that as

P

moves in the interior of triangle

\triangle ABC

, line

\overline{MN}

passes through a fixed point. Proposed by Ali Zamani

geometryIGO

2020 IGO Intermediate P4

Source: 7th Iranian Geometry Olympiad (Intermediate) P4

11/4/2020

Triangle

ABC

is given. An arbitrary circle with center

J

, passing through

B

and

C

, intersects the sides

AC

and

AB

at

E

and

F

, respectively. Let

X

be a point such that triangle

FXB

is similar to triangle

EJC

(with the same order) and the points

X

and

C

lie on the same side of the line

AB

. Similarly, let

Y

be a point such that triangle

EYC

is similar to triangle

FJB

(with the same order) and the points

Y

and

B

lie on the same side of the line

AC

. Prove that the line

XY

passes through the orthocenter of the triangle

ABC

.
Proposed by Nguyen Van Linh - Vietnam

geometryorthocenterIGO

2020 IGO Advanced P4

Source: 7th Iranian Geometry Olympiad (Advanced) P4

11/4/2020

Convex circumscribed quadrilateral

ABCD

with its incenter

I

is given such that its incircle is tangent to

\overline{AD},\overline{DC},\overline{CB},

and

\overline{BA}

at

K,L,M,

and

N

. Lines

\overline{AD}

and

\overline{BC}

meet at

E

and lines

\overline{AB}

and

\overline{CD}

meet at

F

. Let

\overline{KM}

intersects

\overline{AB}

and

\overline{CD}

at

X,Y

, respectively. Let

\overline{LN}

intersects

\overline{AD}

and

\overline{BC}

at

Z,T

, respectively. Prove that the circumcircle of triangle

\triangle XFY

and the circle with diameter

EI

are tangent if and only if the circumcircle of triangle

\triangle TEZ

and the circle with diameter

FI

are tangent. Proposed by Mahdi Etesamifard

geometryIGOiranian geometry olympiadincentertangential quadrilateral

4

Problems(3)