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 $a$ and $b$ ; chord CD splits into $c$ and $d$ at P, so:

a \times b = c \times d

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 $t = ET$ the tangent, $e = EA$ the outer part, and $s = EB$ the whole secant:

t^{2} = e \times s

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: $e_{1} = EA$ , $s_{1} = EB$ . Second secant: $e_{2} = EC$ , $s_{2} = ED$ . Then:

e_{1} \times s_{1} = e_{2} \times s_{2}

Example One from each case

Two chords

One chord splits into 10 and

x

; the other into 5 and 12.

10 \times x = 5 \times 12

, so

10x = 60

⟹ x = 6.

Tangent and secant

The tangent is 8; the secant has outer part

x

and inner part 7, so its whole length is

x + 7

8^{2} = x\,(x + 7)

, so

x^{2} + 7x - 64 = 0

⟹ x ≈ 5.2.

Two secants

First secant: outer 8, inner

x

. Second secant: outer 6, inner 10.

8\,(8 + x) = 6\,(6 + 10) = 96

, so

8 + x = 12

⟹ 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 — $a \times b = c \times d$ .
A tangent and a secant from an outside point: the tangent squared equals the outer part times the whole secant — $t^{2} = e \times s$ .
Two secants from an outside point: outer times whole is the same for both — $e_{1} \times s_{1} = e_{2} \times s_{2}$ .
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.