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