MathDB

Sharygin CR 2019 P6

Source:

3/6/2019

Two quadrilaterals

ABCD

and

A_1B_1C_1D_1

are mutually symmetric with respect to the point

P

. It is known that

A_1BCD

,

AB_1CD

and

ABC_1D

are cyclic quadrilaterals. Prove that the quadrilateral

ABCD_1

is also cyclic

geometrycyclic quadrilateral

Pentagon and lengths

Source: Sharygin 2019 Finals Day 2 Grade 8 P6

7/31/2019

A point

H

lies on the side

AB

of regular polygon

ABCDE

. A circle with center

H

and radius

HE

meets the segments

DE

and

CD

at points

G

and

F

respectively. It is known that

DG=AH

. Prove that

CF=AH

.

geometrySharygin Geometry Olympiad

Vertices from a right-angled triangle

Source: Sharygin 2019 Finals Day 2 Grade 9 P2

7/31/2019

A non-convex polygon has the property that every three consecutive its vertices from a right-angled triangle. Is it true that this polygon has always an angle equal to

90^{\circ}

or to

270^{\circ}

?

angle chasing candidate, angle bisector, median, symmetric, circumcircle

Source: Sharygin 2018 final 10 p6

8/15/2019

Let

AK

and

AT

be the bisector and the median of an acute-angled triangle

ABC

with

AC > AB

. The line

AT

meets the circumcircle of

ABC

at point

D

. Point

F

is the reflection of

K

about

T

. If the angles of

ABC

are known, find the value of angle

FDA

.

geometryangle bisectormediancircumcircleAngle Chasing

6

Problems(4)