Quadrilaterals and centres of triangles
The properties of parallelograms and other quadrilaterals, the mid-point and intercept theorems, and the four centres of a triangle. The triangle centres return in DSE coordinate geometry questions.
What you will learn
- 23.1 Parallelogram: opposite sides equal, opposite angles equal, diagonals bisect each other.
- 23.2 Rectangle, rhombus and square properties (diagonals equal / perpendicular / bisect angles).
- NF 23.3 Conditions for a parallelogram:
- NF 23.4 Simple geometric proofs using these.
- NF 23.5 Mid-point theorem and intercept theorem.
- Logic: "a square has four equal sides, but a figure with four equal sides is not necessarily a square".
- 24.1 A point on an angle bisector is equidistant from the two arms; a point on a perpendicular bisector is equidistant from the two ends (and the converses).
- NF 24.2 Incentre (equidistant from the 3 sides) and circumcentre (equidistant from the 3 vertices).
- NF 24.3 Centroid (medians meet) and orthocentre (altitudes meet).
You need first: Geometry and data basics, Similar triangles
Key terms
| English | 中文 |
|---|---|
| parallelogram, rectangle, rhombus, square, trapezium | 平行四邊形，長方形，菱形，正方形，梯形 |
| diagonal | 對角線 |
| bisect | 平分 |
| mid-point theorem, intercept theorem | 中點定理，截線定理 |
| angle bisector, perpendicular bisector | 角平分線，垂直平分線 |
| median, altitude | 中線，高 |
| incentre, circumcentre, centroid, orthocentre | 內心，外心，形心，垂心 |
Standard steps
Find angles in a parallelogram
- Opposite angles are equal.
- Adjacent angles add to 180° (int. ∠s, AB // DC).
- Write the reason in brackets after each angle you find.
Mid-point theorem (NF)
If M and N are the mid-points of AB and AC, then MN // BC and MN = ½BC.
- Check that both points are mid-points of two sides of the same triangle.
- The segment joining them is parallel to the third side: MN // BC.
- It is half as long as the third side: MN = ½BC. Give the reason (mid-pt. thm.).
Which centre is which (NF)
| Centre | Lines that meet | Special property | Memory hook |
|---|---|---|---|
| Circumcentre | perpendicular bisectors of sides | same distance to the 3 vertices; centre of the circle through them | "circle circumscribed" |
| Incentre | angle bisectors | same distance to the 3 sides; centre of the circle inside | "inside" |
| Centroid | medians (vertex to mid-point of opposite side) | balance point; divides each median 2 : 1 | "centre of mass" |
| Orthocentre | altitudes (vertex perpendicular to opposite side) | "ortho = right angle" |
Right-angled triangle: the circumcentre is the mid-point of the hypotenuse, and the orthocentre is the right-angle vertex.
- Read which lines meet: perpendicular bisectors, angle bisectors, medians or altitudes.
- Name the centre from the table.
- Use its property: equal distances to the vertices (circumcentre) or to the sides (incentre), or the 2 : 1 split of each median (centroid).
Worked examples
Example 1: Find angles in a parallelogram: ABCD is a parallelogram with ∠A = (3x + 10)° and ∠C = (5x − 30)°. Find x and ∠B.
∠A = ∠C (opp. ∠s of //gram)
3x + 10 = 5x − 30
40 = 2x
x = 20
∠A = 70°
∠B = 180° − 70° = 110° (int. ∠s, AD // BC)Example 2: Mid-point theorem (NF): in △ABC, M and N are the mid-points of AB and AC, and BC = 14 cm. Find MN.
MN = ½ × 14 = 7 cm, and MN // BC (mid-pt. thm.).
Example 3: Which centre is which (NF): G is where the medians of △ABC meet, and the median AD is 12 cm. Name G and find AG. Then, in a right-angled triangle with sides 6 cm, 8 cm and 10 cm, find the distance from the circumcentre to each vertex.
medians meet at the centroid, which divides each median 2 : 1
AG = 12 × 2/3 = 8 cm, GD = 4 cm
right-angled triangle: circumcentre = mid-point of the hypotenuse
distance to each vertex = 10 ÷ 2 = 5 cmCommon mistakes
- Treating every quadrilateral with equal sides as a square.
- Mixing up the incentre (sides) and the circumcentre (vertices).
- Thinking the medians are perpendicular to the sides (that's the altitudes).
- Using "properties" as a reason when the proof needs a "condition" (proving a shape is a parallelogram needs a condition).
How the DSE tests it
- A(1)/A(2) proofs: prove a quadrilateral is a parallelogram, or use its properties in a congruence proof.
- Section B: find the circumcentre or orthocentre with coordinates, or show points are concyclic.
- Paper 2: which statement about the centres is true.
Questions you might not ask in class
What is the difference between a property and a condition?
A property: if it is a parallelogram, then this is true. A condition: if this is true, then it is a parallelogram. Same facts, opposite direction.
Can the circumcentre be outside the triangle?
Yes, for an obtuse-angled triangle.
I can't remember four centres.
Remember what meets: perpendicular bisectors → circumcentre, angle bisectors → incentre, medians → centroid, altitudes → orthocentre.