Proposition III.35 — If in a circle two straight lines cut one another, the recta…

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

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

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Depends on (5)

II.5Proposition II.5If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole…
III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at…
I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
1Common notion 1Things which are equal to the same thing are also equal to one another.
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

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