Special Segments in a Circle

Three related theorems describe the lengths of chords, secants, and tangents drawn to a circle. In every case the same idea appears: from a fixed point, the product of the two distances measured along a line through the circle stays constant — a quantity called the power of the point.

Case 1 Two chords inside the circle

When two chords cross inside a circle, the product of the two pieces of one chord equals the product of the two pieces of the other.

Chord AB splits into and ; chord CD splits into and at P, so:

Case 2 A tangent and a secant from outside

From a point outside the circle, the square of the tangent length equals the outer part of the secant times its whole length.

With the tangent, the outer part, and the whole secant:

Case 3 Two secants from outside

From a point outside the circle, each secant's outer part times its whole length gives the same product.

First secant: , . Second secant: , . Then:

Example One from each case

Two chords

One chord splits into 10 and

; the other into 5 and 12.

, so

⟹ x = 6.

Tangent and secant

The tangent is 8; the secant has outer part

and inner part 7, so its whole length is

, so

⟹ x ≈ 5.2.

Two secants

First secant: outer 8, inner

. Second secant: outer 6, inner 10.

, so

⟹ x = 4.

Reference The three rules

Configuration	Where they meet	Rule
Two chords	inside the circle	a × b = c × d
Tangent & secant	outside the circle	t² = e × s
Two secants	outside the circle	e₁ × s₁ = e₂ × s₂

Summary

Two chords inside a circle: the products of their pieces are equal — .
A tangent and a secant from an outside point: the tangent squared equals the outer part times the whole secant — .
Two secants from an outside point: outer times whole is the same for both — .
One idea underlies all three: along any line through a fixed point and the circle, the product of the two distances is constant — the power of the point.