D If CD is a perp. O B O If OC ⊥ AB, then: C C AC = CB A (line from centre ⊥ to chord) x A (perp bisector of chord) C If Ĉ & D̂ are in the same segment & are subtended from the same chord, AB, D If AB is a diameter, O If Ĉ = x, then: B then Ĉ = 90◦ . AÔB = 2x then Ĉ = D̂ B (6 s in same seg) A B (6 at centre = 2 × 6 at circumference.) (6 s in semicircle) C D Given cyclic quad ABCD, From 6 s in same segment: x if D̂ = x, then then the circle’s centre lies on CD C If AC = CB then: C A O B bisector of AB, OC ⊥ AB A (line from centre to midpt of chord) C A B 6 s subtended by equal chords are equal. (equal chords; equal 6 s) B̂ = 180◦ − x If AB subtends equal 6 s D (on the same side of AB), then A, B, C, and D 6 A C (opp 6 s of cyclic quad) B s subtended by equal chords in equal circles are equal. (equal circles; equal chords; equal 6 s) are concyclic. B (converse 6 s in same seg) A D Given cyclic quad ABCD, D Given quad ABCD & D̂ = x x x if D̂ = x, then ◦ If B̂ = 180 − x, then A B̂E = x To prove a quad is cyclic... ABCD is a cyclic quad. 180◦ − x A C (opp 6 s quad supp) B A All four pts are given on the same circle’s circumference Given radius OB and O A tangent AC, then or O B̂C = 90 A B E D Given tangents at A B ABCD is a cyclic quad C (tans from common pt) A B chord BD at B. x then Ê = x C (tan chord theorem) Given line AC touching E touching chord BD with D B̂C = x x In quad ABCD with D̂ = x If A B̂E = x, then A E Given tangent AC of cyclic quad) then BC = AC ◦ C (tan ⊥ radius) E 6 x and B meeting at C, OB ⊥ AC C (ext B x A If D B̂C = Ê then AC is a tangent. B C (converse: tan chord theorem) x B C (ext 6 = int opp 6 ) GR 11 CIRCLES BG