Proof of the
Central Angle Theorem and
April 25, 2006
The study of Euclidean Geometry is based on the ideas of Euclid of Alexandria and his most famous work, The Elements. Euclid is generally thought of as “the father of
geometry” and The Elements is considered one of the most successful textbooks in the
1 The Elements was written over 2,300 years ago and to this day, history of mathematics.
2no copies are still in existence. The Elements is made up of 13 books, each of which serves as a foundation for all who study geometry today. In Euclid’s third book, he
discusses circles, segments of circles, and sectors of circles, including the Central Angle
Theorem and its corollaries.
In order to fully understand the Central Angle Theorem and its corollaries, we must first state a few definitions. If ！？C(O,r)is a circle, an inscribed angle is an angle of the form，PQR, such that P, Q, and R lies on. A central angle is an angle of the ！
form, where O is the center of the circle and P and R are on. An inscribed ，POR！
P angle’s corresponding central angle exists when either “Q and R lie
3on opposite sides of OP or P and Q lie on opposite sides of OR.”
Therefore, a central angle only corresponds to an inscribed angle if Q O
0and only if the inscribed angle has measure less than90. If the
0inscribed angle is greater than 90, it can be seen from the figure
R ，PQRto the right that while inscribed angle intercepts the arc (clockwise from P to PR
R), central angle intercepts a different arc, the arc PQR (counter-clockwise from ，POR
P to R).
1 "Euclid." Wikipedia, The Free Encyclopedia. 20 Apr 2006, 20 Apr 2006
<http://en.wikipedia.org/w/index.php?title=Euclid&oldid=49291859>. 2 Allen, G. Donald. Home page. 17 March 2006. 20 April 2006
<http://www.math.tamu.edu/~don.allen/history/euclid/euclid.html>. 3 Venema, Gerard A., Foundations of Geometry, 239.
The Central Angle Theorem states that, “the measure of an inscribed angle for a
4 In order to prove the circle is one half the measure of the corresponding central angle.”Central Angle Theorem, we begin with the hypothesis that is an inscribed angle ，PQR
in circle. There are three cases to consider. ！？C(O,r)
Case 1: O lies on one of the sides of . (See Fig. 1) To begin, assume Q * ，PQR
O * R (this is equivalent to the case where the angle makes it R * O * Q). To make the
notation simpler, we will let a？，(，OQP) and let b？，(，ORP). Since OQ, OR, and
OP are all radii of the circle and thereby of equal measure, by the Isosceles Triangle Theorem, ，(，OPQ)？，(，OQP)？a and ，(，OPR)？，(，ORP)？b. By the Angle
0((，QPR)？2a;2b？180Sum Theorem, . Also by the Angle Sum Theorem,
0，(，POR)？180；2b and by substitution, . Since the ，(，POR)？2a;2b；2b？2a
inscribed angle ，(，PQR)？a and the corresponding central angle，(，POR)？2a, we
have proven that the measure of the inscribed angle is one half the measure of the corresponding central angle, proving the Central Angle Theorem for this case.
Case 2: O lies in the interior of ，PQR. (See Fig. 2) Construct a point S such
that S lies on and Q * O * S, thereby making S antipodal to Q. By the Protractor ！
Postulate, ，(，PQR)？，(，PQS);，(，RQS) and ，(，POR)？，(，POS);，(，ROS).
Again, to make the notation simpler, we will let a？，(，OQP), b？，(，OSP),
c？，(，OQR)d？，(，OSR), and . Since OQ, OS, OR, and OP are all radii of the circle
and thereby of equal measure, by the Isosceles Triangle Theorem, ，(，OPQ)？，(，OQP)？a，(，OPS)？，(，OSP)？b，(，ORQ)？，(，OQR)？c, , ,
4 Venema, Gerard A., Foundations of Geometry, 239.
. By the Angle Sum Theorem, and ，(，ORS)？，(，OSR)？d
00((，QPS)？2a;2b？180((，QRS)？2c;2d？180 and . Also by the Angle Sum
00，(，POS)？180；2b，(，ROS)？180；2dTheorem, and . Therefore, by substitution,