Angle at center
The measure of the central angle is double the measure of the inscribed angle.
Try to move the points A , B , C And watch the Angle sizes yourself
B
A
C
O
PROOF
Given a circle whose center is point O, choose three points V, C, and D on the circle. Draw lines VC and VD: angle DVC is an inscribed angle. Now draw line VO and extend it past point O so that it intersects the circle at point E. Angle DVC subtends arc DC on the circle.
Suppose this arc includes point E within it. Point E is diametrically opposite to point V. Angles DVE and EVC are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.
∠DVC = ∠DVE + ∠EVC
Then let
ψ0 = ∠DVC , ψ1 = ∠DVE , ψ2 = ∠EVC
So that ψ0 = ψ1 + ψ2
Draw lines OC and OD. Angle DOC is a central angle, but so are angles DOE and EOC, and ∠DOC = ∠DOE + ∠EOC
Then let θ0 = ∠DOC , θ1 = ∠DOE , θ2 = ∠EOC
From Part One we know that
θ1 = 2ψ1 and that θ2 = 2ψ2
Combining these results θ0 = 2ψ1 + 2ψ2 and therefore
θ0 = 2ψ0
Then let
ψ0 = ∠DVC , ψ1 = ∠DVE , ψ2 = ∠EVC
So that ψ0 = ψ1 + ψ2
Draw lines OC and OD. Angle DOC is a central angle, but so are angles DOE and EOC, and ∠DOC = ∠DOE + ∠EOC
Then let θ0 = ∠DOC , θ1 = ∠DOE , θ2 = ∠EOC
From Part One we know that
θ1 = 2ψ1 and that θ2 = 2ψ2
Combining these results θ0 = 2ψ1 + 2ψ2 and therefore
θ0 = 2ψ0