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 centrod 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> | + | In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the centrod 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 == | ||
{{solution}} | {{solution}} | ||
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| + | == See Also == | ||
| + | {{IMO box|year=1964|num-b=5|after=Last Question}} | ||
| + | [[Category:Olympiad Geometry Problems]] | ||
| + | [[Category:3D Geometry Problems]] | ||
Revision as of 12:50, 29 January 2021
Problem
In tetrahedron 
, vertex 
is connected with 
, the centrod 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 | ||
