Proposition III.20 — In a circle the angle at the centre is double of the angle a…

Proof

Let

∠ B E C

be the angle at the centre

E

and

∠ B A C

the angle at the circumference, both subtending the arc

B C

. Join

A E

and produce to

F

on the far side of the circle. In

△ E A B

E A = E B

(radii), so by I.5,

∠ E A B = ∠ E B A

. By I.32 (exterior angle of a triangle),

∠ B E F = ∠ E A B + ∠ E B A = 2 \cdot ∠ E A B

. Similarly in

△ E A C

∠ C E F = 2 \cdot ∠ E A C

. Adding (Common Notion 2):

∠ B E C = ∠ B E F + ∠ C E F = 2 \cdot (∠ E A B + ∠ E A C) = 2 \cdot ∠ B A C

. (Heath handles the case where

A

lies on the arc opposite

B C

from

E

via a subtraction instead of an addition; the result is the same.)

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Depends on (5)

I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal straight lines be produced…
I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
2Common notion 2If equals be added to equals, the wholes are equal.
3Common notion 3If equals be subtracted from equals, the remainders are equal.
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) (6)

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