Question

Use vector methods to prove that the lines joining the mid-points of the opposite edges of a tetrahedron \(O A B C\) meet at a point and that this point bisects each of the lines.

Short answer

Lines joining the midpoints intersect at \ \mathbf{R} \ and this point bisects each line.

Step-by-step solution

- Define vectors and midpoints

Let the position vectors of points O, A, B, and C be denoted by \(\mathbf{O} = \mathbf{0}\), \(\mathbf{A} = \mathbf{a} \, \, \mathbf{B} = \mathbf{b} \, \, \mathbf{C} = \mathbf{c}\). The midpoints of the edges are \(M_1\) (between \( \mathbf{O} \) and \( \mathbf{A} \)), \(M_2\) (between \( \mathbf{B} \) and \( \mathbf{C} \)), \(N_1\) (between \( \mathbf{O} \) and \( \mathbf{B} \)), \(N_2\) (between \( \mathbf{A} \) and \( \mathbf{C} \)), etc.

- Calculate the midpoints

Calculate the midpoints: \[ \mathbf{M_1} = \frac{\mathbf{O} + \mathbf{A}}{2} = \frac{\mathbf{a}}{2} \] \[ \mathbf{M_2} = \frac{\mathbf{B} + \mathbf{C}}{2} \] \[ \mathbf{N_1} = \frac{\mathbf{O} + \mathbf{B}}{2} = \frac{\mathbf{b}}{2} \] \[ \mathbf{N_2} = \frac{\mathbf{A} + \mathbf{C}}{2} \]

- Form equations of lines

Write the parametric equations of lines joining midpoints: Line joining \(\mathbf{M_1} \) and \(\mathbf{M_2} \): \[ \mathbf{r_1} = \mathbf{M_1} + t(\mathbf{M_2} - \mathbf{M_1}) = \frac{\mathbf{a}}{2} + t\left( \frac{\mathbf{b} + \mathbf{c}}{2} - \frac{\mathbf{a}}{2} \right) \] Line joining \(\mathbf{N_1} \) and \(\mathbf{N_2} \): \[ \mathbf{r_2} = \mathbf{N_1} + s(\mathbf{N_2} - \mathbf{N_1}) = \frac{\mathbf{b}}{2} + s\left( \frac{\mathbf{a} + \mathbf{c}}{2} - \frac{\mathbf{b}}{2} \right) \]

- Find intersection of lines

Set the equations equal and solve for t and s: \ \frac{\mathbf{a}}{2} + t \left( \frac{\mathbf{b} + \mathbf{c} - \mathbf{a}}{2} \right) = \frac{\mathbf{b}}{2} + s \left( \frac{\mathbf{a} + \mathbf{c} - \mathbf{b}}{2} \right). Equate coefficients and solve for \( t \) and \( s \).

- Verify the intersection point

Substitute values of t and s back into equations. Both should give the same vector \( \mathbf{R} \). This point is the midpoint of the lines joining the tetrahedron's midpoints.

Key concepts

Position Vectors

Position vectors make locating points in space much easier. In the context of a tetrahedron, we often define these position vectors from a common origin.
For example, consider a tetrahedron with vertices O, A, B, and C. The position vector of O is \(\textbf{O} = \textbf{0}\), A is \(\textbf{A} = \textbf{a}\), B is \(\textbf{B} = \textbf{b}\), and C is \(\textbf{C} = \textbf{c}\).
This method helps simplify calculations significantly.
Whenever we need to refer to a location, we just look at the vector indicating its position relative to the origin.

Midpoints

When dealing with the midpoints of a line segment, there's a simple formula we use to find their position vectors.
Suppose we want the midpoint of the line segment joining two points with position vectors \(\textbf{P}\) and \(\textbf{Q}\). The midpoint, \(\textbf{M}\), has a position vector of: \[ \textbf{M} = \frac{\textbf{P} + \textbf{Q}}{2} \]
For our tetrahedron, we have midpoints between various vertices: \(\textbf{M}_1\) (between O and A), \(\textbf{M}_2\) (between B and C), \(\textbf{N}_1\) (between O and B), \(\textbf{N}_2\) (between A and C).
Using the formula, we calculate these midpoints as follows:

• \(\textbf{M}_1 = \frac{\textbf{a}}{2} \)
• \(\textbf{M}_2 = \frac{\textbf{b} + \textbf{c}}{2} \)
• \(\textbf{N}_1 = \frac{\textbf{b}}{2} \)
• \(\textbf{N}_2 = \frac{\textbf{a} + \textbf{c}}{2} \)

Parametric Equations

A parametric equation represents a line in space by expressing its coordinates as functions of a parameter, usually denoted by \(t\) or \(s\). This approach is particularly useful when we want to describe a line passing through given points.
For the tetrahedron's midpoints, let’s write the parametric equations of lines joining \(\textbf{M}_1\) & \(\textbf{M}_2\), and \(\textbf{N}_1\) & \(\textbf{N}_2\):

• Line from \(\textbf{M}_1\) to \(\textbf{M}_2\):
  \[ \textbf{r}_1 = \textbf{M}_1 + t(\textbf{M}_2 - \textbf{M}_1) = \frac{\textbf{a}}{2} + t\bigg(\frac{\textbf{b} + \textbf{c}}{2} - \frac{\textbf{a}}{2}\bigg) \]
• Line from \(\textbf{N}_1\) to \(\textbf{N}_2\):
  \[ \textbf{r}_2 = \textbf{N}_1 + s(\textbf{N}_2 - \textbf{N}_1) = \frac{\textbf{b}}{2} + s\bigg(\frac{\textbf{a} + \textbf{c}}{2} - \frac{\textbf{b}}{2}\bigg) \]

These equations help us track points on the lines as \(t\) and \(s\) vary.

Intersection of Lines

To prove that the lines intersect, we equate their parametric equations and solve for \(t\) and \(s\).
From the equations:
\( \frac{\textbf{a}}{2} + t \bigg( \frac{\textbf{b} + \textbf{c} - \textbf{a}}{2} \bigg) = \frac{\textbf{b}}{2} + s \bigg( \frac{\textbf{a} + \textbf{c} - \textbf{b}}{2} \bigg) \)
We solve for \(t\) and \(s\) by equating coefficients. This ensures the two points on each line are the same. By substituting these values back into the equations, we find the intersecting point or verify that both lines genuinely intersect.
This intersecting point's position vector will be the average, confirming that it indeed bisects the lines joining midpoints. Therefore, the intersection point is both the midpoint and the common point where the lines meet.