Difference between revisions of "1964 IMO Problems/Problem 6"
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== Problem == | == Problem == | ||
− | In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the | + | In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the centroid of <math>\triangle ABC</math>. Lines parallel to <math>DD_0</math> are drawn through <math>A,B</math> and <math>C</math>. These lines intersect the planes <math>BCD, CAD</math> and <math>ABD</math> in points <math>A_1, B_1,</math> and <math>C_1</math>, respectively. Prove that the volume of <math>ABCD</math> is one third the volume of <math>A_1B_1C_1D_0</math>. Is the result true if point <math>D_o</math> is selected anywhere within <math>\triangle ABC</math>? |
== Solution == | == Solution == |
Revision as of 23:16, 16 November 2023
Problem
In tetrahedron , vertex
is connected with
, the centroid of
. Lines parallel to
are drawn through
and
. These lines intersect the planes
and
in points
and
, respectively. Prove that the volume of
is one third the volume of
. Is the result true if point
is selected anywhere within
?
Solution
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See Also
1964 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Question |
All IMO Problems and Solutions |