MathDB

Squares drawn outwardly of ABC

Source: Federation of Bosnia, 1. Grades 2008.

4/23/2008

Squares

BCA_{1}A_{2}

,

CAB_{1}B_{2}

,

ABC_{1}C_{2}

are outwardly drawn on sides of triangle

\triangle ABC

. If

AB_{1}A'C_{2}

,

BC_{1}B'A_{2}

,

CA_{1}C'B_{2}

are parallelograms then prove that: (i) Lines

BC

and

AA'

are orthogonal. (ii)Triangles

\triangle ABC

and

\triangle A'B'C'

have common centroid

geometryparallelogramanalytic geometryvectorcomplex numbers

Nice geometry

Source: Federation of Bosnia and Heryegovina, 2nd grades, 2008.

4/28/2008

Given is an acute angled triangle

\triangle ABC

with side lengths

a

,

b

and

c

(in an usual way) and circumcenter

O

. Angle bisector of angle

\angle BAC

intersects circumcircle at points

A

and

A_{1}

. Let

D

be projection of point

A_{1}

onto line

AB

,

L

and

M

be midpoints of

AC

and

AB

, respectively. (i) Prove that AD\equal{}\frac{1}{2}(b\plus{}c) (ii) If triangle

\triangle ABC

is an acute angled prove that A_{1}D\equal{}OM\plus{}OL

geometrycircumcirclerectangletrigonometryangle bisector

Nice and old one

Source: Federation of Bosnia and Heryegovina, 3rd grades, 2008.

4/28/2008

Two circles with centers

S_{1}

and

S_{2}

are externally tangent at point

K

. These circles are also internally tangent to circle

S

at points

A_{1}

and

A_{2}

, respectively. Denote by

P

one of the intersection points of

S

and common tangent to

S_{1}

and

S_{2}

at

K

.Line

PA_{1}

intersects

S_{1}

at

B_{1}

while

PA_{2}

intersects

S_{2}

at

B_{2}

. Prove that

B_{1}B_{2}

is common tangent of circles

S_{1}

and

S_{2}

.

Really nice

Source: Federation of Bosnia and Herzegovina, 4th grades, 2008.

5/1/2008

Given are three pairwise externally tangent circles

K_{1}

,

K_{2}

and

K_{3}

. denote by

P_{1}

tangent point of

K_{2}

and

K_{3}

and by

P_{2}

tangent point of

K_{1}

and

K_{3}

. Let

AB

(

A

and

B

are different from tangency points) be a diameter of circle

K_{3}

. Line

AP_{2}

intersects circle

K_{1}

(for second time) at point

X

and line

BP_{1}

intersects circle

K_{2}

(for second time) at

Y

. If

Z

is intersection point of lines

AP_{1}

and

BP_{2}

prove that points

X

,

Y

and

Z

are collinear.

trigonometrygeometrycircumcirclegeometry proposed

1

Problems(4)