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1996 Cabri Clubs 1st , Round 2, 6 problems, Argentinian geo contest

Source:

November 25, 2021

geometryconstructioncabri clubsgeometric constructionLocus

Problem Statement

level 1
p1. Given a circle

\omega

and two points

A

and

M

on

\omega

. Take a point

B

on

\omega

and construct a point

C

such that the segment

AM

is the median of

ABC

. Find

B

so that the area of triangle

ABC

is maximum.
p2. A triangle

ABC

and a point

P

are given. Let

S_1

be the symmetric point of

P

wrt

A

,

S_2

the symmetric point of

S_1

wrt

B

and

S_3

the symmetric point of

S_2

wrt

C

. Find the locus of

S_3

as

P

moves.
p3. Given a segment

AB

and a point

X

on

AB

. The line

r

, perpendicular to

AB

, is drawn at

X

. Let

O

be a point on the line

r

and

\omega

the circle with center

O

passing through

A

. The tangents to

\omega

are traced through

B

, touching

\omega

at

T_1

and

T_2

. Find the locus of

T_1

and

T_2

as

O

moves on

r

.
level 2
p4. Given a circle

\omega

and a chord

DA

on the circle. Take another point

B

on

\omega

(different from

A

and

D

) and draw the line

s

passing through

A

perpendicular on

DB

. The intersection point of line

s

and line

DB

is

C

. Find point

B

so that the area of triangle

ABC

is maximum.
p5. Given a triangle

ABC

, let

P

be any point in the plane. The perpendicular bisectors of

AP

,

BP

and

CP

that intersect at three points,

A'

,

B'

and

C'

, are drawn. Find a point

P

so that the triangle

ABC

and

A'B'C'

are similar.
p6. Let

C_1

and

C_2

be two intersecting circles at

A

and

B

where

AB

is the diameter of

C_1

. Let

P

be a point on

C_2

, in the interior of

C_1

. Construct two points

C

and

D

on

C_1

such that

CD

is perpendicular to

AB

and

\angle CPD

is right, using only the constructions of:

\bullet

a line given by two points

\bullet

intersection of

2

objects

\bullet

line perpendicular on a line.

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