Proposition IV.3 — About a given circle to circumscribe a triangle equiangular …

Proof

Let

A B C

be the given circle with centre

K

and

D E F

the given triangle. Produce

E F

both ways to

G

H

. At the centre

K

construct

∠ A K B = ∠ D E G

and

∠ A K C = ∠ D F H

(I.23). At

A

B

C

draw the tangents to the circle (III.16, III.17); they bound a triangle. Because each tangent is perpendicular to the radius at the point of contact (III.18), the angles of the constructed triangle are the supplements of the central angles

∠ A K B

∠ A K C

∠ B K C

, hence equal to the angles of

△ D E F

by I.13 and the construction.

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Depends on (5)

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.23Proposition I.23On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
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,…
III.16Proposition III.16The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle,…
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…

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