Proposition III.9 — If a point be taken within a circle, and more than two equal…

Proof

Let

F

be the point and

F A

F B

F C

three equal lines to the circle. Join

A B

B C

; bisect them at

G

H

(I.10). Join

F G

F H

. In

△ F A G

and

△ F B G

F A = F B

given,

A G = GB

by construction,

F G

common; by I.8 the triangles are congruent, so

∠ F G A = ∠ F GB

, and by I.13 both are right. Similarly

F H ⊥ B C

. By III.3 (rewriting it as: the perpendicular at the midpoint of a chord passes through the centre), both

F G

produced and

F H

produced pass through the centre. Their intersection

F

is therefore the centre.

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Depends on (6)

III.1Proposition III.1To find the centre of a given circle.
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.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they…
I.10Proposition I.10To bisect a given finite straight line.
I.13Proposition I.13If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two…
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) (2)

III.10Proposition III.10A circle does not cut a circle at more points than two.
III.25Proposition III.25Given a segment of a circle, to describe the complete circle of which it is a segment.

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