MathDB

On Perpendiculars and Collinearity

Source: Indonesian Stage 1 TST for IMO 2022, Test 1 (Geometry)

12/11/2021

Given an acute triangle

ABC

. with

H

as its orthocenter, lines

\ell_1

and

\ell_2

go through

H

and are perpendicular to each other. Line

\ell_1

cuts

BC

and the extension of

AB

on

D

and

Z

respectively. Whereas line

\ell_2

cuts

BC

and the extension of

AC

on

E

and

X

respectively. If the line through

D

and parallel to

AC

and the line through

E

parallel to

AB

intersects at

Y

, prove that

X,Y,Z

are collinear.

geometryorthocentercollinearperpendicularExtensionacuteTriangle

Generalization of a Midpoint Theorem

Source: Indonesian Stage 1 TST for IMO 2022, Test 2 (Geometry)

12/23/2021

Given that

ABC

is a triangle, points

A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})

and

O_A, O_B, O_C

satisfy the following criteria:
a)

ABB_1A_2, BCC_1B_2, CAA_1C_2

are rectangles not containing any interior points of the triangle

ABC

, b)

\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}

, c)

AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2

are parallelograms, and d)

O_A

is the centroid of rectangle

BCC_1B_2

,

O_B

is the centroid of rectangle

CAA_1C_2

, and

O_C

is the centroid of rectangle

ABB_1A_2

.
Prove that

A_3O_A, B_3O_B,

and

C_3O_C

concur at a point.
Proposed by Farras Mohammad Hibban Faddila

Trianglegeometrymidpointconcurrence

On concyclicity (Indonesia Stage 1 TST for IMO 2022)

Source: Indonesian Stage 1 TST for IMO 2022, Test 3 (Geometry)

12/25/2021

Let

AB

be the diameter of circle

\Gamma

centred at

O

. Point

C

lies on ray

\overrightarrow{AB}

. The line through

C

cuts circle

\Gamma

at

D

and

E

, with point

D

being closer to

C

than

E

is.

OF

is the diameter of the circumcircle of triangle

BOD

. Next, construct

CF

, cutting the circumcircle of triangle

BOD

at

G

. Prove that

O,A,E,G

are concyclic.
(Possibly proposed by Pak Wono)

geometrydiametercircumcircleConcyclic

Configurational Incenter-Excenter Geometry

Source: Indonesian Stage 1 TST for IMO 2022, Test 4 (Geometry)

12/25/2021

In a nonisosceles triangle

ABC

, point

I

is its incentre and

\Gamma

is its circumcircle. Points

E

and

D

lie on

\Gamma

and the circumcircle of triangle

BIC

respectively such that

AE

and

ID

are both perpendicular to

BC

. Let

M

be the midpoint of

BC

,

N

be the midpoint of arc

BC

on

\Gamma

containing

A

,

F

is the point of tangency of the

A-

excircle on

BC

, and

G

is the intersection of line

DE

with

\Gamma

. Prove that lines

GM

and

NF

intersect at a point located on

\Gamma

.
(Possibly proposed by Farras Faddila)

Fact 5geometry

G

Problems(4)