MathDB

Possible 12-Gon Construction?

Source: Sharygin Geometry Olympiad 2014 - Problem 2

11/15/2014

A paper square with sidelength

2

is given. From this square, can we cut out a

12

-gon having all sidelengths equal to

1

and all angles divisible by

45^\circ

?

geometry unsolvedgeometry

in a quadrilateral with 2 opposite angles right ... a line bisects a segment

Source: 2014 Sharygin Geometry Olympiad Final Round 9.2

8/3/2018

In a quadrilateral

ABCD

angles

A

and

C

are right. Two circles with diameters

AB

and

CD

meet at points

X

and

Y

. Prove that line

XY

passes through the midpoint of

AC

.
(F. Nilov )

circlesgeometry

segments of feet of altitudes // segment of feet of angle biectors, isosceles ?

Source: 2014 Sharygin Geometry Olympiad Final Round 8.2

8/3/2018

Let

AH_a

and

BH_b

be altitudes,

AL_a

and

BL_b

be angle bisectors of a triangle

ABC

. It is known that

H_aH_b // L_aL_b

. Is it necessarily true that

AC = BC

?
(B. Frenkin)

geometryisosceles

intesecting tangents to create a segment of constant length

Source: 2014 Sharygin Geometry Olympiad Final Round 10.2

8/3/2018

A circle, its chord

AB

and the midpoint

W

of the minor arc

AB

are given. Take an arbitrary point

C

on the major arc

AB

. The tangent to the circle at

C

meets the tangents at

A

and

B

at points

X

and

Y

respectively. Lines

WX

and WY meet AB at points

N

and

M

respectively. Prove that the length of segment

NM

does not depend on point

C

.
(A. Zertsalov, D. Skrobot)

geometrytangentconstantSegment

2

Problems(4)