Circles & Circle Theorems

At a glance 11 formulas

1. 1Chord and centre$\left(\tfrac{\text{chord}}{2}\right)^2 + d^2 = r^2$$r = \text{radius},\ d = \text{centre to chord}$
2. 2Tangent length$\text{tangent} = \sqrt{d^2 - r^2}$$r = \text{radius},\ d = \text{point to centre}$
3. 3Equal tangentsThe two tangents from an outside point are equal. So an incircle cuts each side of a triangle into tangent lengths, and a quadrilateral with an incircle has $AB + CD = BC + DA$.
4. 4Chords, secants through P$PA \times PB = PC \times PD$$\text{tangent } PT:\ PT^2 = PA \times PB$
5. 5Tangents, centres D apart$\text{direct} = \sqrt{D^2 - (r_1 - r_2)^2}$$\text{transverse} = \sqrt{D^2 - (r_1 + r_2)^2}$
6. 6Angle at the centre$\angle AOB = 2\angle ACB;\ \text{semicircle: } 90^\circ$$O = \text{centre},\ C \text{ on the major arc}$
7. 7Same segmentAngles that stand on the same arc, from points on the same side of its chord, are equal.
8. 8Cyclic quadrilateral$\angle A + \angle C = \angle B + \angle D = 180^\circ$$\text{exterior angle} = \text{interior opposite angle}$
9. 9Alternate segmentThe angle between a tangent and a chord at the point of contact equals the angle that chord makes at the circle on the other side.
10. 10Regions and sectorsJoin the centre to where the circle crosses the figure, then add or subtract triangles and sectors ($\frac{\theta}{360} \pi r^2$ for angle $\theta$, radius $r$).
11. 11Right-triangle incircle$\text{area} = BD \times DC$$\angle A = 90^\circ,\ \text{incircle touches } BC \text{ at } D$