Key theorems relating arcs, chords, inscribed angles, and tangent lines in circles. Covers the Inscribed Angle Theorem (an inscribed angle is half its intercepted arc), Thales' theorem (inscribed angle in a semicircle is 90 degrees), tangent-radius perpendicularity, and the two-tangent theorem.
High School Advanced • 9-12
Circles are arguably the most elegant objects in geometry, and the theorems governing them are among the most beautiful results in all of mathematics. In this lesson, you will explore the deep relationships between arcs, chords, central angles, and inscribed angles. These theorems have been known since the time of Euclid and Thales, and they remain essential in modern mathematics, engineering, and physics.
| Term | Definition |
|---|---|
| Central Angle | An angle whose vertex is the center of the circle. Its measure equals the measure of its intercepted arc. |
| Inscribed Angle | An angle whose vertex is on the circle and whose sides are chords of the circle. |
| Intercepted Arc | The arc that lies in the interior of an angle and has endpoints on the angle's sides. |
| Chord | A segment whose endpoints are both on the circle. |
| Tangent | A line that touches the circle at exactly one point (the point of tangency). |
| Secant | A line that intersects the circle at two points. |
A central angle and its intercepted arc have the same measure in degrees. This is actually the definition of arc measure. From this starting point:
This is the crown jewel of circle theorems:
Equivalently, an inscribed angle is half the central angle that subtends the same arc. If the intercepted arc measures 80°, the inscribed angle measures 40°.
Corollaries of the Inscribed Angle Theorem:
If AB is a diameter of a circle and C is any point on the circle (other than A or B), then angle ACB = 90°.
Why? The arc AB intercepted by the inscribed angle ACB is a semicircle (180°). By the Inscribed Angle Theorem, the inscribed angle is half of 180° = 90°.
This theorem is attributed to Thales of Miletus (circa 600 BCE) and is considered one of the first proven theorems in history.
Tangent-Radius Perpendicularity: A tangent line to a circle is perpendicular to the radius drawn to the point of tangency. This gives us a right angle every time a tangent meets a radius.
Two-Tangent Theorem: If two tangent lines are drawn to a circle from the same external point, then:
Problem: In circle O, central angle AOB = 110°. Point C is on the major arc AB. Find the measure of inscribed angle ACB.
Solution:
The central angle AOB intercepts minor arc AB, which measures 110°.
The inscribed angle ACB also intercepts minor arc AB (since C is on the major arc).
By the Inscribed Angle Theorem:
Problem: AB is a diameter of circle O with radius 5. Point C is on the circle such that AC = 6. Find BC.
Solution:
By Thales' Theorem, angle ACB = 90° (angle inscribed in a semicircle).
The diameter AB = 2 × 5 = 10.
By the Pythagorean Theorem in right triangle ACB:
Problem: From external point P, two tangent lines are drawn to circle O (radius 5), touching at points A and B. If PA = 12, find the distance from P to center O.
Solution:
By the tangent-radius theorem, OA is perpendicular to PA, creating right angle OAP.
In right triangle OAP: OA = 5 (radius), PA = 12 (tangent segment).
Note: By the Two-Tangent Theorem, PB = PA = 12 as well.
When a quadrilateral is inscribed in a circle (all four vertices on the circle), opposite angles are supplementary. This means if you know three angles of an inscribed quadrilateral, the fourth is determined. This property is unique to cyclic quadrilaterals and can be used to prove that a quadrilateral is or is not inscribable in a circle.
Problem 1: In circle O, inscribed angle ABC intercepts arc AC = 140°. Find m∠ABC.
By the Inscribed Angle Theorem: m∠ABC = (1/2)(140°) = 70°.
Problem 2: Quadrilateral PQRS is inscribed in a circle. If ∠P = 85° and ∠Q = 110°, find ∠R and ∠S.
Opposite angles of an inscribed quadrilateral are supplementary.
∠R = 180° - ∠P = 180° - 85° = 95°
∠S = 180° - ∠Q = 180° - 110° = 70°
Check: 85 + 110 + 95 + 70 = 360°. Confirmed.
Problem 3: AB is a diameter of a circle with radius 13. C is a point on the circle with BC = 10. Find AC.
By Thales' Theorem, ∠ACB = 90°. AB = 26 (diameter).
AC² + BC² = AB²
AC² + 100 = 676
AC² = 576, so AC = 24.
Problem 4: From an external point P, a tangent PA and a secant PBC are drawn to circle O. If PA = 8 and PB = 4, find PC. (Use the tangent-secant theorem: PA² = PB × PC.)
By the tangent-secant theorem: PA² = PB × PC
64 = 4 × PC
PC = 16
So the chord BC has length PC - PB = 16 - 4 = 12.
Problem 5: Two chords AB and CD intersect inside circle O at point E. If AE = 3, EB = 8, and CE = 4, find ED. (Use the intersecting chords theorem: AE × EB = CE × ED.)
By the intersecting chords theorem: AE × EB = CE × ED
3 × 8 = 4 × ED
24 = 4 × ED
ED = 6
Circle theorems reveal the elegant internal structure of circles. The Inscribed Angle Theorem -- that an inscribed angle equals half its intercepted arc -- is the foundation from which Thales' Theorem and the inscribed quadrilateral property follow. The tangent-radius perpendicularity gives you right angles whenever a tangent meets a radius, and the Two-Tangent Theorem ensures equal tangent segments from any external point. These theorems work together to solve problems involving arcs, chords, angles, and tangent lines.