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Czech-Polish-Slovak Match
2004 Czech-Polish-Slovak Match
5
5
Part of
2004 Czech-Polish-Slovak Match
Problems
(1)
Triangles that share an orthocentre
Source: Czech-Polish-Slovak 2004 Q5
4/28/2013
Points
K
,
L
,
M
K,L,M
K
,
L
,
M
on the sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
respectively of a triangle
A
B
C
ABC
A
BC
satisfy
A
K
K
B
=
B
L
L
C
=
C
M
M
A
\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}
K
B
A
K
=
L
C
B
L
=
M
A
CM
. Show that the triangles
A
B
C
ABC
A
BC
and
K
L
M
KLM
K
L
M
have a common orthocenter if and only if
△
A
B
C
\triangle ABC
△
A
BC
is equilateral.
geometry
vector
circumcircle
geometry unsolved