Centroids, collinear points and shapes

What you will be able to do

• find the centroid of a triangle and of a tetrahedron;
• decide whether three points are collinear;
• recognise a parallelogram, a rectangle, a rhombus, a square and a right triangle from the coordinates of the corners.

The centre of a flat plate

Hang a triangular plate from a corner and it balances at one special point, the centroid. For a triangle with corners $A$, $B$, $C$ the centroid $G$ is the mean of the corners:

It lies on each median, a line from a corner to the midpoint of the opposite side, and it divides every median in the ratio $2 : 1$ from the corner. You can check this with the section formula. The midpoint $M$ of $BC$ is $\dfrac{B + C}{2}$. The point that divides $AM$ in the ratio $2 : 1$ is $\dfrac{2M + A}{3} = \dfrac{A + B + C}{3}$.

For a tetrahedron with corners $A$, $B$, $C$, $D$ the centroid is the mean of the four corners. It divides the line from a corner to the centroid of the opposite face in the ratio $3 : 1$.

Example. The triangle $A(1, 2, 3)$, $B(4, -2, 6)$, $C(-2, 5, 3)$ has $G = \left(\dfrac{3}{3}, \dfrac{5}{3}, \dfrac{12}{3}\right) = \left(1, \dfrac{5}{3}, 4\right)$. Add $D(2, -4, 4)$ and the tetrahedron $ABCD$ has centroid $\left(\dfrac{5}{4}, \dfrac{1}{4}, 4\right)$.

Three points on one line

Points $A$, $B$, $C$ are collinear if they lie on one line. Two tests work.

1. By distances. If $AB + BC = AC$, then $B$ lies between $A$ and $C$ on a line. Use the actual lengths, not squares. For $(1, 2, 3)$, $(3, 5, 7)$ and $(5, 8, 11)$, the differences are $(2, 3, 4)$ each time. So $AB = BC = \sqrt{29}$ and $AC = 2\sqrt{29}$. They are collinear.
2. By the ratio. Find the ratio in which $B$ divides $AC$ from each coordinate. If all three agree, the points are collinear.

If the test fails, the three points are the corners of a triangle.

Shapes from their corners

For a quadrilateral $ABCD$, take the corners in order round the shape.

• Parallelogram. The diagonals $AC$ and $BD$ have the same midpoint. In coordinates $A + C = B + D$. So if $A$, $B$, $C$ are known, then $D = A + C - B$.
• Rectangle. A parallelogram with equal diagonals.
• Rhombus. A parallelogram with two adjacent sides equal.
• Square. A parallelogram with equal adjacent sides and equal diagonals.
• Right triangle. The largest squared side is the sum of the other two.

Lengths in space do not need a picture. Compare squared lengths of sides and of diagonals.

Example. The quadrilateral $A(1, 0, 2)$, $B(3, 1, 4)$, $C(7, -3, 2)$, $D(5, -4, 0)$ has $A + C = (8, -3, 4)$ and $B + D = (8, -3, 4)$, so it is a parallelogram. The squared diagonals are $AC^2 = 36 + 9 + 0 = 45$ and $BD^2 = 4 + 25 + 16 = 45$. They are equal, so it is a rectangle. Its sides $AB = 3$ and $BC = 6$ differ, so it is not a square.

Practise with new numbers

Try it in Python

The function classify uses squared lengths to name a quadrilateral. The program tests three shapes.

Common mistakes

• Dividing by the wrong number. The centroid of a triangle divides by $3$ and of a tetrahedron by $4$.
• Listing the corners in the wrong order. For $ABCD$ the diagonals are $AC$ and $BD$.
• Calling a parallelogram a rectangle because the angles look right. Test the diagonals or a dot product, never the picture.
• Treating equal diagonals as enough. Equal diagonals make a rectangle only if the quadrilateral is already a parallelogram.

Remember these

• Centroid of a triangle: the mean of 3 corners. Of a tetrahedron: the mean of 4. It cuts a median in the ratio $2 : 1$ and a tetrahedron line in $3 : 1$.
• Collinear: $AB + BC = AC$, or one section ratio from all three coordinates.
• Parallelogram: $A + C = B + D$. Rectangle: also $AC = BD$. Rhombus: also $AB = BC$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the centroid of the triangle with corners $(2, -1, 3)$, $(5, 4, 0)$ and $(-1, 6, 6)$.

2. Find the centroid of the tetrahedron with corners $(0, 0, 0)$, $(4, 0, 0)$, $(0, 8, 0)$ and $(0, 0, 12)$.

3. $ABCD$ is a parallelogram with $A(1, 0, 2)$, $B(4, 2, 1)$ and $C(7, 5, 4)$. Find $D$.

4. Are $(1, 2, 3)$, $(3, 5, 7)$ and $(5, 8, 11)$ collinear? Give a reason.

5. Is the parallelogram $A(1, 0, 2)$, $B(3, 1, 4)$, $C(2, 3, 6)$, $D(0, 2, 4)$ a rhombus, a rectangle or neither?

References

• NCERT, Mathematics Textbook for Class XI, the chapter Introduction to Three Dimensional Geometry, examples on the centroid and on collinear points.
• OpenStax, Calculus Volume 3, Chapter 2, section 2.2 on vectors in three dimensions.