MathDB

isosceles trapezoid wanted when all polygons are regular

Source: 2023 IGO Elementary P1

1/18/2024

All of the polygons in the figure below are regular. Prove that

ABCD

is an isosceles trapezoid. https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png
Proposed by Mahdi Etesamifard - Iran

geometrytrapezoidisosceles

IGO 2023 Intermidiate P1

Source: IGO 2023 Intermidiate P1

1/18/2024

Points

M

and

N

are the midpoints of sides

AB

and

BC

of the square

ABCD

. According to the fgure, we have drawn a regular hexagon and a regular

12

-gon. The points

P, Q

and

R

are the centers of these three polygons. Prove that

PQRS

is a cyclic quadrilateral.
Proposed by Mahdi Etesamifard - Iran

geometry

4 concyclic wanted, 5 given

Source: 2023 IGO Adcanced P1

1/18/2024

We are given an acute triangle

ABC

. The angle bisector of

\angle BAC

cuts

BC

at

P

. Points

D

and

E

lie on segments

AB

and

AC

, respectively, so that

BC \parallel DE

. Points

K

and

L

lie on segments

PD

and

PE

, respectively, so that points

A

,

D

,

E

,

K

,

L

are concyclic. Prove that points

B

,

C

,

K

,

L

are also concyclic.
Proposed by Patrik Bak, Slovakia

geometryConcyclic

1

Problems(3)