1964 IMO Problems/Problem 6

Problem

In tetrahedron $ABCD$ , vertex $D$ is connected with $D_0$ , the centroid of $\triangle ABC$ . Lines parallel to $DD_0$ are drawn through $A,B$ and $C$ . These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1,$ and $C_1$ , respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$ . Is the result true if point $D_o$ is selected anywhere within $\triangle ABC$ ?

Solution

Let $A_{2}$ be the point where line $AD_{0}$ intersects line $BC$

Let $B_{2}$ be the point where line $BD_{0}$ intersects line $AC$

Let $C_{2}$ be the point where line $CD_{0}$ intersects line $AB$

From centroid properties we have:

$|AA_{2}|=3|D_{0}A_{2}|$

$|BB_{2}|=3|D_{0}B_{2}|$

$|CC_{2}|=3|D_{0}C_{2}|$

Since $\Delta D_{0}A_{2}A_{1}\sim \Delta AA_{2}A_{1}$ , then $|AA_{2}|=|DD_{0}| \frac{|AA_{2}|}{|D_{0}A_{2}|}

This problem needs a solution. If you have a solution for it, please help us out by adding it.

1964 IMO (Problems) • Resources
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6	Followed by Last Question
All IMO Problems and Solutions

Page

Toolbox

Search

1964 IMO Problems/Problem 6

Problem

Solution

See Also