Source — Euclid's Elements, encoded as an rrxiv paper

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III.35 Proposition III.35

If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

Proof

Let chords

A B

and

C D

meet at

E

inside the circle with centre

O

. Let

M

be the midpoint of

A B

and

N

of

C D

; by III.3,

O M ⊥ A B

and

O N ⊥ C D

. Set

r =

radius. Apply II.5 to chord

A B

bisected at

M

and cut at

E

:

A E \cdot E B + E M^{2} = A M^{2}

. By I.47 in right

△ O M A

,

A M^{2} = r^{2} - O M^{2}

. Substituting:

A E \cdot E B = r^{2} - O M^{2} - E M^{2} = r^{2} - O E^{2}

(using

O E^{2} = O M^{2} + E M^{2}

, I.47 in right

△ O M E

). So

A E \cdot E B = r^{2} - O E^{2}

, depending only on the distance

O E

from the centre. The same identity applies to chord

C D

:

C E \cdot E D = r^{2} - O E^{2}

. Hence

A E \cdot E B = C E \cdot E D

.

lines 74–74 in main.tex