Lesson 2 — Cyclic Quadrilaterals
Big idea: A quadrilateral can be inscribed in a circle exactly when a pair of its opposite angles add to 180°.
Words to know
| Word | Meaning |
|---|---|
| Cyclic quadrilateral | A quadrilateral whose vertices all lie on one circle. |
| Opposite angles | Angles of a quadrilateral that do not share a side: ∠A and ∠C, ∠B and ∠D. |
| Supplementary | Two angles whose measures add to 180°. |
| Inscribed angle | An angle whose vertex is on a circle and whose sides are chords. |
| Isosceles trapezoid | A trapezoid whose legs are congruent. |
| Ptolemy’s Theorem | In a cyclic quadrilateral, the sum of the products of the opposite sides equals the product of the diagonals. |
Concepts to know
- Theorem 69: A quadrilateral is cyclic iff a pair of its opposite angles are supplementary.
- Forward proof: an inscribed angle is half its intercepted arc. The two opposite arcs make the whole circle (360°), so the opposite angles add to 180°.
- The other pair of opposite angles also adds to 180°, because a quadrilateral’s angles total 360°.
- Converse: if a pair of opposite angles is supplementary, the fourth vertex must lie on the circle through the other three.
- A rectangle is cyclic because each angle is 90°.
- An isosceles trapezoid is cyclic.
- Unlike triangles, not every quadrilateral is cyclic.
- From the exercises: Ptolemy’s Theorem and Brahmagupta’s area formula apply to cyclic quadrilaterals.
Formulas / rules
- Cyclic iff ∠A + ∠C = 180° and ∠B + ∠D = 180°.
- Ptolemy: AB · DC + AD · BC = AC · BD.
- Brahmagupta: A = √((s − a)(s − b)(s − c)(s − d)), where s = ½(a + b + c + d).
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Decide whether a quadrilateral is cyclic | Add a pair of opposite angles; it is cyclic iff the sum is 180°. |
| Find a missing angle | Subtract the opposite angle from 180°. |
| Use an intercepted arc | Double the inscribed angle, or halve the arc. |
| Work with an isosceles trapezoid | Use the fact that it is always cyclic. |
| Work with a rectangle | Its diagonals are diameters of the circumcircle. |
| Apply Ptolemy | Label the sides in order around the circle before multiplying. |
Watch out
- Supplementary means 180°, not 90°.
- Use opposite angles, not consecutive angles.
- If one pair of opposite angles is supplementary, the other pair automatically is too.
- Ptolemy and Brahmagupta apply only to cyclic quadrilaterals.
- A quadrilateral can have all sides equal and still not be cyclic; a rhombus usually is not.