Circles — CDS Elementary Mathematics

Formulas (2)

• Half-chord, distance and radius · Chord and distance
  $$r^2 = d^2 + \left(\dfrac{c}{2}\right)^2$$
• Arches, segment heights and equal chords · Radius from a segment
  $$r = \dfrac{(c/2)^2 + h^2}{2h}$$

Watch out for (4)

• Parallel chords have two answers
  → Half-chord, distance and radius
• Half the chord, not the chord
  → Half-chord, distance and radius
• The centre is r − h from the chord
  → Arches, segment heights and equal chords
• Radius or diameter?
  → Arches, segment heights and equal chords

Formulas (2)

• Angle at the centre and angles in the same segment · Inscribed angle
  $$\angle AOB = 2\,\angle ACB$$
• Cyclic quadrilaterals and the two segments · Cyclic quadrilateral
  $$\angle A + \angle C = \angle B + \angle D = 180^\circ$$

Watch out for (3)

• An obtuse inscribed angle needs the reflex angle
  → Angle at the centre and angles in the same segment
• Two diameters meet at the centre
  → Angle at the centre and angles in the same segment
• Segment, not sector
  → Cyclic quadrilaterals and the two segments

Formulas (2)

• Circumcircle of a triangle · Circumradius
  $$R = \dfrac{abc}{4\Delta}$$
• Locus of a moving point · Sum of squares
  $$PA^2 + PB^2 = 2\,PM^2 + \dfrac{AB^2}{2}$$

Watch out for (2)

• Equal distance from every vertex means circumcentre
  → Circumcircle of a triangle
• Sum of squares is a circle, not a line
  → Locus of a moving point

Formulas (3)

• Length of a tangent · Tangent length
  $$PT = \sqrt{OP^2 - r^2}$$
• The alternate segment theorem · Tangent–chord angle
  $$\angle QPT = \angle PRQ = \tfrac12\angle POQ$$
• A circle inside a quadrilateral · Tangential quadrilateral
  $$AB + CD = BC + DA$$

Watch out for (3)

• Supplement, not double
  → Length of a tangent
• The ALTERNATE segment
  → The alternate segment theorem
• Opposite sides, not adjacent ones
  → A circle inside a quadrilateral

Formulas (1)

• Chords, secants and the tangent from one point · Tangent and secant
  $$PT^2 = PA \cdot PB$$

Watch out for (2)

• The whole secant, then subtract
  → Chords, secants and the tangent from one point
• Given the product, the statements are not needed
  → Chords, secants and the tangent from one point

Formulas (2)

• Direct and transverse common tangents · Common tangents
  $$\text{direct} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad \text{transverse} = \sqrt{d^2 - (r_1 + r_2)^2}$$
• The common chord of intersecting circles · Distance between centres
  $$d = \sqrt{r_1^2 - h^2} + \sqrt{r_2^2 - h^2}$$

Watch out for (2)

• Difference for direct, sum for transverse
  → Direct and transverse common tangents
• Both centres can be on one side
  → The common chord of intersecting circles

Formulas (2)

• Radii from contact and areas · Difference of the radii
  $$(r_1 - r_2)^2 = 2(r_1^2 + r_2^2) - (r_1 + r_2)^2$$
• Circles inside an angle · Two circles in an angle
  $$\dfrac{r_2}{r_1} = \dfrac{1 - \sin\theta}{1 + \sin\theta}$$

Watch out for (3)

• Internal contact uses the difference
  → Radii from contact and areas
• Diameters, not radii
  → Radii from contact and areas
• Sine of the HALF-angle
  → Circles inside an angle