Proposition III.36 — If a point be taken outside a circle and from it there fall …

Proof

Let

P

be the external point,

P T

the tangent at

T

, and

P A B

a secant cutting the circle at

A

(near) and

B

(far); let

O

be the centre and

r

the radius. By III.18,

O T ⊥ P T

. By I.47 in

△ O T P

O P^{2} = O T^{2} + P T^{2} = r^{2} + P T^{2}

, hence

P T^{2} = O P^{2} - r^{2}

. Let

M

be the midpoint of

A B

; by III.3,

O M ⊥ A B

. Apply II.6 to

A B

bisected at

M

and produced to

P

(so

P

is on line

A B

extended beyond the near intersection

A

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

. By I.47 in

△ O M A

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

; in

△ O M P

M P^{2} = O P^{2} - O M^{2}

. Subtracting:

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

. Comparing:

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

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Depends on (7)

II.5Proposition II.5If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole…
II.6Proposition II.6If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the…
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…
III.18Proposition III.18If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight…
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…

Required by (dependents) (1)

III.37Proposition III.37If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut…

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