Difference between revisions of "1964 IMO Problems/Problem 6"
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<math>\frac{h_{\Delta A_{1}B_{1}C_{1}}}{h_{D}}=\frac{|AA_{2}|}{|D_{0}A_{2}|}=\frac{|BB_{2}|}{|D_{0}B_{2}|}=\frac{|CC_{2}|}{|D_{0}C_{2}|}=3</math> | <math>\frac{h_{\Delta A_{1}B_{1}C_{1}}}{h_{D}}=\frac{|AA_{2}|}{|D_{0}A_{2}|}=\frac{|BB_{2}|}{|D_{0}B_{2}|}=\frac{|CC_{2}|}{|D_{0}C_{2}|}=3</math> | ||
− | <math> | + | <math>h_{\Delta A_{1}B_{1}C_{1}}=3h_{D}</math> |
Revision as of 23:02, 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
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 |