MathDB

level 1
p1. i. Given three not collinear points

A

B

and

C

, find points

A'

B'

and

C'

BC

AC

and

AB

, respectively, such that the triangles

ABC

and

A'B'C'

have the same centroid.
ii. Given three not collinear points

A

B

and

C

, find points

A'

B'

and

C'

BC

AC

and

AB

, respectively, such that the triangles

ABC

and

A'B'C'

have the same circumcenter.
p2. Given a parallelogram

ABCD

, find four points

A'

B'

C'

and

D'

all interior to the parallelogram such that the triples of points

AA'D'

BB'A'

CC'B'

and

DD'C'

are collinear and the polygons

ADD'

DC'C

CB'B

AA'B

and

A'B'C'D'

have equal areas.
p3. Construct a circle with center

O

. Let

AB

and

SJ

be two perpendicular diameters, in the minor arc

BS

take a variable point

H

. For each

H

we determine

F

, the intersection point between

AH

and

SB

and

T

the intersection point between

SA

and

BH

. i. Show that

TF

and

AB

are orthogonal. ii. Find the locus of the circumcenter of

FSH

.
level 2
p4. Given a triangle

ABC

, find the locus of the points

P

inside the triangle such that

d (P,BC) = d (P,AC) + d (P,AB)

.
Note:

d (P, BC)

is the distance from

P

from line

BC

.
p5. Construct a circle and take a fixed point

A

on it. Draw a variable line

r

through

A

that is secant to the circle at

A

and

B

. Construct an isosceles trapezoid

ABCD

of bases

AB

and

CD

such that

AD = DC = 1/2 AB

. i. Show that line

BC

passes through a fixed point. ii. Find the locus of

C

. iii. A line

t

perpendicular to

AC

is drawn through

D

. Let

P

be the intersection point between

t

and

AB

. Take

J

on the ray

AB

and

X

on the ray

PD

such that

AJ = PX

. Show that the perpendicular bisector of

JX

passes through the incenter of triangle

APD

.
p6. Construct a triangle

ABC

, and then find points

A'

B'

and

C'

BC

AC

and

AB

, respectively, such that

A'B'

is perpendicular to

BC

B'C'

is perpendicular to

AC

and

C'A'

is perpendicular to

AB

Problem Statement