Proposition III.30 — To bisect a given arc.

Proof

Let

A D B

be the given arc with chord

A B

. Bisect

A B

C

(I.10). At

C

erect

C D ⊥ A B

(I.11), meeting the arc at

D

. Join

A D

B D

. In right triangles

△ A C D

and

△ B C D

A C = C B

(construction),

C D

common, right angles at

C

. By I.4, the triangles are congruent and

A D = B D

. By III.28, equal chords cut off equal arcs (in the same circle), so arc

A D

equals arc

B D

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Depends on (4)

III.28Proposition III.28In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to…
I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
I.10Proposition I.10To bisect a given finite straight line.
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.

Required by (dependents) (2)

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