Proposition IV.2 — In a given circle to inscribe a triangle equiangular with a …

Proof

Let

A B C

be the given circle and

D E F

the given triangle. Draw the tangent

G H

at any point

A

of the circle (III.16, III.17). On

G H

construct

∠ H A C = ∠ D E F

(I.23), and

∠ G A B = ∠ D F E

. Join

B C

. By III.32 (tangent–chord angle equals the inscribed angle in the alternate segment),

∠ A C B = ∠ H A B = ∠ D E F

and

∠ A B C = ∠ G A C = ∠ D F E

. By I.32 the remaining angles agree; so

△ A B C

is equiangular with

△ D E F

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Full neighborhood

Depends on (4)

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.32Proposition III.32If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line…

Required by (dependents) (4)

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