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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1998 Chile National Olympiad
2
2
Part of
1998 Chile National Olympiad
Problems
(1)
locus wanted, semicircle, constant length (1998 Chile Level 2 P2)
Source:
6/17/2020
Given a semicircle of diameter
A
B
AB
A
B
, with
A
B
=
2
r
AB = 2r
A
B
=
2
r
, be
C
D
CD
C
D
a variable string, but of fixed length
c
c
c
. Let
E
E
E
be the intersection point of lines
A
C
AC
A
C
and
B
D
BD
B
D
, and let
F
F
F
be the intersection point of lines
A
D
AD
A
D
and
B
C
BC
BC
. a) Prove that the lines
E
F
EF
EF
and
A
B
AB
A
B
are perpendicular. b) Determine the locus of the point
E
E
E
. c) Prove that
E
F
EF
EF
has a constant measure, and determine it based on
c
c
c
and
r
r
r
.
geometry
Locus
semicircle