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National and Regional Contests
Chile Contests
Chile National Olympiad
1992 Chile National Olympiad
5
5
Part of
1992 Chile National Olympiad
Problems
(1)
computational in a triangle (Chile 1992 L2 P5)
Source:
5/27/2019
In the
△
A
B
C
\triangle ABC
△
A
BC
, points
M
,
I
,
H
M, I, H
M
,
I
,
H
are feet, respectively, of the median, bisector and height, drawn from
A
A
A
. It is known that
B
C
=
2
BC = 2
BC
=
2
,
M
I
=
2
−
3
MI = 2-\sqrt {3}
M
I
=
2
−
3
and
A
B
>
A
C
AB > AC
A
B
>
A
C
. a) Prove that
I
I
I
lies between
M
M
M
and
H
H
H
. b) Calculate
A
B
2
−
A
C
2
AB ^ 2-AC ^ 2
A
B
2
−
A
C
2
. c) Determine
A
B
A
C
\dfrac {AB} {AC}
A
C
A
B
. d) Find the measure of all the sides and angles of the triangle.
geometry
angles
Sides