1964 IMO Problems/Problem 6
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
Since and
then,
thus,
this proves that the volume of is one third the volume of
The result is NOT true if point is selected anywhere within as ratios of , , and will have values other than 3 as the point is no longer a centroid. Also, it will make the ratios which means that and the volume relationship will no longer hold true.
~Tomas Diaz. orders@tomasdiaz.com
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 |
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