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National and Regional Contests
France Contests
French Mathematical Olympiad
1991 French Mathematical Olympiad
Problem 3
Problem 3
Part of
1991 French Mathematical Olympiad
Problems
(1)
inscribed tetrahedra with orthogonal segments
Source: France 1991 P3
5/14/2021
Let
S
S
S
be a fixed point on a sphere
Σ
\Sigma
Σ
with center
Ω
\Omega
Ω
. Consider all tetrahedra
S
A
B
C
SABC
S
A
BC
inscribed in
Σ
\Sigma
Σ
such that
S
A
,
S
B
,
S
C
SA,SB,SC
S
A
,
SB
,
SC
are pairwise orthogonal. (a) Prove that all the planes
A
B
C
ABC
A
BC
pass through a single point. (b) In one such tetrahedron,
H
H
H
and
O
O
O
are the orthogonal projections of
S
S
S
and
Ω
\Omega
Ω
onto the plane
A
B
C
ABC
A
BC
, respectively. Let
R
R
R
denote the circumradius of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that
R
2
=
O
H
2
+
2
S
H
2
R^2=OH^2+2SH^2
R
2
=
O
H
2
+
2
S
H
2
.
3D geometry
geometry