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National and Regional Contests
Poland Contests
Poland - Second Round
1985 Poland - Second Round
6
6
Part of
1985 Poland - Second Round
Problems
(1)
right angle and coplanar points wanted
Source: Polish MO Recond Round 1985 p6
9/9/2024
There are various points in space
A
,
B
,
C
0
,
C
1
,
C
2
A, B, C_0, C_1, C_2
A
,
B
,
C
0
,
C
1
,
C
2
, with
∣
A
C
i
∣
=
2
∣
B
C
i
∣
|AC_i| = 2 |BC_i|
∣
A
C
i
∣
=
2∣
B
C
i
∣
for
i
=
0
,
1
,
2
i = 0,1,2
i
=
0
,
1
,
2
and
∣
C
1
C
2
∣
=
4
3
∣
A
B
∣
|C_1C_2|=\frac{4}{3}|AB|
∣
C
1
C
2
∣
=
3
4
∣
A
B
∣
. Prove that the angle
C
1
C
0
C
2
C_1C_0C_2
C
1
C
0
C
2
is right and the points
A
,
B
,
C
1
,
C
2
A, B, C_1, C_2
A
,
B
,
C
1
,
C
2
lie on one plane.
geometry
3D geometry
right angle