Circles and circle theorems

Coordinate geometry of the circle (MM3.2)

• \(x^2 + y^2 + cx + dy + e = 0\) is \((x + \frac c2)^2 + (y + \frac d2)^2 = \frac{c^2}{4} + \frac{d^2}{4} - e\); it is a real circle only when the right-hand side is positive.
• Tangent at a point: perpendicular to the radius there. Length of the tangent from an external point \(P\): \(\sqrt{PC^2 - r^2}\).
• A line is a tangent iff its distance from the centre equals \(r\) (or the substituted quadratic has discriminant \(0\)).
• Two circles with radii \(r_1, r_2\) and centres \(d\) apart: touch externally if \(d = r_1 + r_2\), internally if \(d = |r_1 - r_2|\), intersect twice if \(|r_1 - r_2| < d < r_1 + r_2\).
• A right angle in a triangle inscribed in a circle sits opposite a diameter, so the hypotenuse is a diameter and its midpoint is the centre.

Circle theorems (MM3.3, M5.9)

• The perpendicular from the centre to a chord bisects the chord.
• The tangent is perpendicular to the radius at the point of contact.
• Angle at the centre \(= 2 \times\) angle at the circumference (same arc).
• The angle in a semicircle is \(90^\circ\).
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
• Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.