Difference between revisions of "1964 IMO Problems/Problem 6"
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Since <math>\Delta D_{0}A_{2}A_{1}\sim \Delta AA_{2}A_{1}</math>, then <math>|AA_{2}|=|DD_{0}| \frac{|AA_{2}|}{|D_{0}A_{2}|}=3|DD_{0}|</math> | Since <math>\Delta D_{0}A_{2}A_{1}\sim \Delta AA_{2}A_{1}</math>, then <math>|AA_{2}|=|DD_{0}| \frac{|AA_{2}|}{|D_{0}A_{2}|}=3|DD_{0}|</math> | ||
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+ | Since <math>\Delta D_{0}B_{2}B_{1}\sim \Delta BB_{2}B_{1}</math>, then <math>|BB_{2}|=|DD_{0}| \frac{|BB_{2}|}{|D_{0}B_{2}|}=3|DD_{0}|</math> | ||
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+ | Since <math>\Delta D_{0}C_{2}C_{1}\sim \Delta CC_{2}C_{1}</math>, then <math>|CC_{2}|=|DD_{0}| \frac{|CC_{2}|}{|D_{0}C_{2}|}=3|DD_{0}|</math> | ||
Revision as of 22:50, 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
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
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 |