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line perpendicular to the median of a triangle

Source: Sharygin Geometry Olympiad 2015 Final 8.6

8/1/2018

Lines

b

and

c

passing through vertices

B

and

C

of triangle

ABC

are perpendicular to sideline

BC

. The perpendicular bisectors to

AC

and

AB

meet

b

and

c

at points

P

and

Q

respectively. Prove that line

PQ

is perpendicular to median

AM

of triangle

ABC

.
(D. Prokopenko)

geometryperpendicular bisectorperpendiculargeometry solvedSharygin Geometry OlympiadRussiamedian

in a convex ABCD with perpendicular diagonals, 4 lines are concurrent

Source: Sharygin Geometry Olympiad 2015 Final 9.6

8/1/2018

The diagonals of convex quadrilateral

ABCD

are perpendicular. Points

A' , B' , C' , D'

are the circumcenters of triangles

ABD, BCA, CDB, DAC

respectively. Prove that lines

AA' , BB' , CC' , DD'

concur.
(A. Zaslavsky)

geometryconcurrencyconcurrentdiagonals

Circumcircle geometry

Source: Sharygin geometry olympiad 2015, grade 10, Final Round, Problem 6

7/17/2018

Let

H

and

O

be the orthocenter and the circumcenter of triangle

ABC

. The circumcircle of triangle

AOH

meets the perpendicular bisector of

BC

at point

A_1 \neq O

. Points

B_1

and

C_1

are defined similarly. Prove that lines

AA_1

,

BB_1

and

CC_1

concur.

geometrycircumcircleperpendicular bisector

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