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Polish MO Finals
1978 Polish MO Finals
6
sum 1/ h_i^2 = sum 1/ d_i^2 in tetrahedron
sum 1/ h_i^2 = sum 1/ d_i^2 in tetrahedron
Source: Polish MO Finals 1978 p6
August 24, 2024
geometry
3D geometry
tetrahedron
Problem Statement
Prove that if
h
1
,
h
2
,
h
3
,
h
4
h_1,h_2,h_3,h_4
h
1
,
h
2
,
h
3
,
h
4
are the altitudes of a tetrahedron and
d
1
,
d
2
,
d
3
d_1,d_2,d_3
d
1
,
d
2
,
d
3
the distances between the pairs of opposite edges of the tetrahedron, then
1
h
1
2
+
1
h
2
2
+
1
h
3
2
+
1
h
4
2
=
1
d
1
2
+
1
d
2
2
+
1
d
3
2
.
\frac{1}{h_1^2} +\frac{1}{h_2^2} +\frac{1}{h_3^2} +\frac{1}{h_4^2} =\frac{1}{d_1^2} +\frac{1}{d_2^2} +\frac{1}{d_3^2}.
h
1
2
1
+
h
2
2
1
+
h
3
2
1
+
h
4
2
1
=
d
1
2
1
+
d
2
2
1
+
d
3
2
1
.
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