MathDB

2

criterion for a convex hexagon with parallel opposite sides to be cyclic

Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle.
(A. Zaslavsky)

PQ passes through circumcenter, similar triangles of projections of 2 points,

Given a triangle

ABC

and points

P

and

Q

. It is known that the triangles formed by the projections

P

and

Q

on the sides of

ABC

are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line

PQ

passes through the center of the circumscribed circle of triangle

ABC

.
(A. Zaslavsky)

5

2

an acute triangle conctruction, given two midpoints and orthocenter

Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides.
(A. Zaslavsky)

polyhedron pasted over with these two shawls without overlap

There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov).

4

2

concyclic projections wanted, inside a cyclic quadrilateral

A circle can be circumscribed around the quadrilateral

ABCD

. Point

P

is the foot of the perpendicular drawn from point

A

on line

BC

, and respectively

Q

from

A

on

DC

,

R

from

D

on

AB

and

T

from

D

on

BC

. Prove that points

P,Q,R

and

T

lie on the same circle.
(A. Myakishev)

distance form circumcenter to orthocenter is AB+ AC when <A=120^o

Angle

A

in triangle

ABC

is equal to

120^o

. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to

AB + AC

.
(V. Protasov)

3

2

angle chasing in a regular hexagon, 45^o given

In the regular hexagon

ABCDEF

on the line

AF

, the point

X

is taken so that the angle

XCD

is

45^o

. Find the angle

\angle FXE

.
(Kiev Olympiad)

is it parallelogram? midpoints, equal segments

Given a quadrilateral

ABCD

.

A ', B', C'

and

D'

are the midpoints of the sides

BC, CB, BA

and

AB

, respectively. It is known that

AA'= CC'

,

BB'= DD'

. Is it true that

ABCD

is a parallelogram?
(M. Volchkevich)

2

inverse of isosceles criterion? external angle bisectors, circumcircle related

In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?

area between 4 common tangents of 2 circles, internal are perpendicular

The radii

r

and

R

of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangents.

1

2

given points (1,2), (3,1) restore coordinate system

A coordinate system was drawn on the board and points

A (1,2)

and

B (3,1)

were marked. The coordinate system was erased. Restore it by the two marked points.

cutting 2 similar triangles into 2 parts, 1 similar with the starting one

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?
(D. Shnol)

2008 Oral Moscow Geometry Olympiad

Subcontests

criterion for a convex hexagon with parallel opposite sides to be cyclic

PQ passes through circumcenter, similar triangles of projections of 2 points,

an acute triangle conctruction, given two midpoints and orthocenter

polyhedron pasted over with these two shawls without overlap

concyclic projections wanted, inside a cyclic quadrilateral

distance form circumcenter to orthocenter is AB+ AC when <A=120^o

angle chasing in a regular hexagon, 45^o given

is it parallelogram? midpoints, equal segments

inverse of isosceles criterion? external angle bisectors, circumcircle related

area between 4 common tangents of 2 circles, internal are perpendicular

given points (1,2), (3,1) restore coordinate system

cutting 2 similar triangles into 2 parts, 1 similar with the starting one

2008 Oral Moscow Geometry Olympiad

Subcontests

criterion for a convex hexagon with parallel opposite sides to be cyclic

PQ passes through circumcenter, similar triangles of projections of 2 points,

an acute triangle conctruction, given two midpoints and orthocenter

polyhedron pasted over with these two shawls without overlap

concyclic projections wanted, inside a cyclic quadrilateral

distance form circumcenter to orthocenter is AB+ AC when &lt;A=120^o

angle chasing in a regular hexagon, 45^o given

is it parallelogram? midpoints, equal segments

inverse of isosceles criterion? external angle bisectors, circumcircle related

area between 4 common tangents of 2 circles, internal are perpendicular

given points (1,2), (3,1) restore coordinate system

cutting 2 similar triangles into 2 parts, 1 similar with the starting one

distance form circumcenter to orthocenter is AB+ AC when <A=120^o