Proposition IV.10 — To construct an isosceles triangle having each of the angles…

Proof

Take a straight line

A B

and cut it at

C

so that the rectangle on

A B

B C

equals the square on

A C

(II.11, the golden section). With centre

A

and radius

A B

describe a circle; in it apply chord

B D

equal to

A C

(IV.1). Join

A D

C D

. Because

A B \cdot B C = A C^{2} = B D^{2}

B D

is tangent to the circle through

A

C

D

(III.37); by III.32 the tangent–chord angle equals the alternate inscribed angle. Tracking the resulting angle relations (with I.5 for the isosceles base angles) gives the required ratio.

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Depends on (7)

II.11Proposition II.11To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the…
IV.1Proposition IV.1Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the…
IV.5Proposition IV.5About a given triangle to circumscribe a circle.
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,…
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…
III.37Proposition III.37If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut…

Required by (dependents) (3)

IV.11Proposition IV.11In a given circle to inscribe an equilateral and equiangular pentagon.
XIII.8Proposition XIII.8If in an equilateral and equiangular pentagon straight lines subtend two adjacent angles, they cut one another in…
XIII.9Proposition XIII.9If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight…

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