Proof
Apply I.3 to mark equal segments either side of the given point, then
I.1 to erect an equilateral triangle whose apex line is perpendicular
(by I.8 and Definition I.10).
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Depends on (4)
- I.1Proposition I.1On a given finite straight line to construct an equilateral triangle.
- I.3Proposition I.3Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
- I.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they…
- I.10Definition I.10When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles…
Required by (dependents) (22)
- 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.46Proposition I.46On a given straight line to describe a square.
- I.48Proposition I.48If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the…
- II.1Proposition II.1If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by…
- II.9Proposition II.9If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double…
- II.10Proposition II.10If a straight line be bisected and a straight line be added to it in a straight line, the square on the whole with the…
- 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…
- II.14Proposition II.14To construct a square equal to a given rectilineal figure.
- III.1Proposition III.1To find the centre of a given circle.
- III.4Proposition III.4If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another.
- 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.17Proposition III.17From a given point to draw a straight line touching a given 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…
- III.19Proposition III.19If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the…
- III.25Proposition III.25Given a segment of a circle, to describe the complete circle of which it is a segment.
- III.30Proposition III.30To bisect a given arc.
- III.33Proposition III.33On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.
- IV.5Proposition IV.5About a given triangle to circumscribe a circle.
- IV.6Proposition IV.6In a given circle to inscribe a square.
- IV.7Proposition IV.7About a given circle to circumscribe a square.
- VI.13Proposition VI.13To two given straight lines to find a mean proportional.
- XI.11Proposition XI.11From a given elevated point to draw a straight line perpendicular to a given plane.
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