Difference between revisions of "1964 IMO Problems/Problem 6"
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<math>h_{\Delta A_{1}B_{1}C_{1}}=3h_{D}</math> | <math>h_{\Delta A_{1}B_{1}C_{1}}=3h_{D}</math> | ||
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+ | <math>h_{\Delta A_{1}B_{1}C_{1}}Area_{\Delta A_{1}B_{1}C_{1}}=3h_{D}Area_{\Delta ABC}</math> | ||
Revision as of 00:04, 17 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
Let be the point where line
intersects line
Let be the point where line
intersects line
Let be the point where line
intersects line
From centroid properties we have:
Therefore,
Since , then
Since , then
Since , then
Since and
,
then , and
Let be the perpendicular distance from
to
Let be the perpendicular distance from
to
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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 |