In equal circles angles standing on equal circumferences are equal to one another
, whether they stand at the centres or at the circumferences
For in equal circles ABC
, on equal circumferences BC
, let the angles BGC
stand at the centres G
, and the angles BAC
at the circumferences; I say that the angle BGC
is equal to the angle EHF
, and the angle BAC
is equal to the angle EDF
For, if the angle BGC
is unequal to the angle EHF
, one of them is greater.
Let the angle BGC
be greater : and on the straight line BG
, and at the point G
on it, let the angle BGK
be constructed equal to the angle EHF
. [I. 23
Now equal angles stand on equal circumferences, when they are at the centres; [III. 26
] therefore the circumference BK is equal to the circumference EF.
is equal to BC
; therefore BK is also equal to BC, the less to the greater : which is impossible.
Therefore the angle BGC
is not unequal to the angle EHF
; therefore it is equal to it.
And the angle at A
is half of the angle BGC
, and the angle at D half of the angle EHF; [III. 20]
therefore the angle at A
is also equal to the angle at D
Therefore etc. Q. E. D.