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Canada Contests
Canada National Olympiad
2024 Canada National Olympiad
5
5
Part of
2024 Canada National Olympiad
Problems
(1)
Constructing orthocenter using ruler with width
Source: Canada MO 2024/5
3/8/2024
Initially, three non-collinear points,
A
A
A
,
B
B
B
, and
C
C
C
, are marked on the plane. You have a pencil and a double-edged ruler of width
1
1
1
. Using them, you may perform the following operations:[*]Mark an arbitrary point in the plane. [*]Mark an arbitrary point on an already drawn line. [*]If two points
P
1
P_1
P
1
and
P
2
P_2
P
2
are marked, draw the line connecting
P
1
P_1
P
1
and
P
2
P_2
P
2
. [*]If two non-parallel lines
l
1
l_1
l
1
and
l
2
l_2
l
2
are drawn, mark the intersection of
l
1
l_1
l
1
and
l
2
l_2
l
2
. [*]If a line
l
l
l
is drawn, draw a line parallel to
l
l
l
that is at distance
1
1
1
away from
l
l
l
(note that two such lines may be drawn). Prove that it is possible to mark the orthocenter of
A
B
C
ABC
A
BC
using these operations.
geometry