Tangent-Chord Theorem


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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 $\angle DBF$.


In the words of Euclid:

If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle.

(The Elements: Book $\text{III}$: Proposition $32$)


Proof

Draw $BA$ perpendicular to $EF$ through $B$.

Let $C$ be selected on the circle on the arc $BD$.

Join $AD, DC, CB$.

From Right Angle to Tangent of Circle goes through Center, the center of the circle lies on $AB$.

By definition, then, $AB$ is a diameter of the circle.

From Relative Sizes of Angles in Segments, it follows that $\angle ADB$ is a right angle.

Therefore from Sum of Angles of Triangle Equals Two Right Angles $\angle BAD + \angle ABD$ equals a right angle.

But $\angle ABF$ is also a right angle.

So $\angle ABF = \angle BAD + \angle ABD$

Subtracting $\angle ABD$ from each, it follows that $\angle DBF = \angle BAD$.

$\Box$


Next we have that $ABCD$ is a cyclic quadrilateral.

From Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles $\angle BAD + \angle BCD$ equals two right angles.

But from Two Angles on Straight Line make Two Right Angles, so does $\angle DBE + \angle DBF$.

But as $\angle BAD = \angle DBF$ it follows that $\angle BCD = \angle DBE$.

$\blacksquare$


Also known as

The is also known as the alternate segment theorem.


Historical Note

This proof is Proposition $32$ of Book $\text{III}$ of Euclid's The Elements.


Sources

  • 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{III}$. Propositions
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circle $(7)$
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tangent-chord theorem
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): circle theorems
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): circle theorems
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