Circle Theorems
Theorem
Inscribed Angle Theorem
An inscribed angle is equal to half the angle that is subtended by that arc.
Thus, in the figure above:
- $\angle ABC = \frac 1 2 \angle ADC$
In the words of Euclid:
- In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.
(The Elements: Book $\text{III}$: Proposition $20$)
Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles
In the words of Euclid:
- The opposite angles of quadrilaterals in circles are equal to two right angles.
(The Elements: Book $\text{III}$: Proposition $22$)
Thales' Theorem
Let $A$ and $B$ be two points on opposite ends of the diameter of a circle.
Let $C$ be another point on the circle such that $C \ne A, B$.
Then the lines $AC$ and $BC$ are perpendicular to each other.
Tangent-Chord Theorem
Let $EF$ be a tangent to a circle $ABCD$, touching it at $B$.
Let $BD$ be a chord of $ABCD$.
Then:
- the angle in segment $BCD$ equals $\angle DBE$
and:
- the angle in segment $BAD$ equals $\angle DBF$.
Tangent Secant Theorem
Let $D$ be a point outside a circle $ABC$.
Let $DB$ be tangent to the circle $ABC$.
Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$.
Then $DB^2 = AD \cdot DC$.
In the words of Euclid:
- If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the tangent.
(The Elements: Book $\text{III}$: Proposition $36$)