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Source — Euclid's Elements, encoded as an rrxiv paper

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Claims in this paper

465 addressable claims. Click to open the claim view with its full neighborhood.

I.1Proposition I.1On a given finite straight line to construct an equilateral triangle.
I.2Proposition I.2To place at a given point (as an extremity) a straight line equal to a given straight…
I.3Proposition I.3Given two unequal straight lines, to cut off from the greater a straight line equal to…
I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles…
I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal…
I.6Proposition I.6If in a triangle two angles are equal to one another, the sides which subtend the equal…
I.7Proposition I.7Given two straight lines constructed on a straight line and meeting in a point, there…
I.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the…
I.9Proposition I.9To bisect a given rectilineal angle.
I.10Proposition I.10To bisect a given finite straight line.
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
I.12Proposition I.12To draw a perpendicular straight line to a given infinite straight line from a given…
I.13Proposition I.13If a straight line set up on a straight line make angles, it will make either two right…
I.14Proposition I.14If with any straight line, and at a point on it, two straight lines not lying on the same…
I.15Proposition I.15If two straight lines cut one another, they make the vertical angles equal to one another.
I.16Proposition I.16In any triangle, if one of the sides be produced, the exterior angle is greater than…
I.17Proposition I.17In any triangle two angles taken together in any manner are less than two right angles.
I.18Proposition I.18In any triangle the greater side subtends the greater angle.
I.19Proposition I.19In any triangle the greater angle is subtended by the greater side.
I.20Proposition I.20In any triangle two sides taken together in any manner are greater than the remaining one.
I.21Proposition I.21If on one of the sides of a triangle, from its extremities, there be constructed two…
I.22Proposition I.22Out of three straight lines, which are equal to three given straight lines, to construct…
I.23Proposition I.23On a given straight line and at a point on it to construct a rectilineal angle equal to a…
I.24Proposition I.24If two triangles have the two sides equal to two sides respectively, but have the one of…
I.25Proposition I.25If two triangles have the two sides equal to two sides respectively, but have the one…
I.26Proposition I.26If two triangles have the two angles equal to two angles respectively, and one side equal…
I.27Proposition I.27If a straight line falling on two straight lines make the alternate angles equal to one…
I.28Proposition I.28If a straight line falling on two straight lines make the exterior angle equal to the…
I.29Proposition I.29A straight line falling on parallel straight lines makes the alternate angles equal to…
I.30Proposition I.30Straight lines parallel to the same straight line are also parallel to one another.
I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two…
I.33Proposition I.33The straight lines joining equal and parallel straight lines (at the extremities which…
I.34Proposition I.34In parallelogrammic areas the opposite sides and angles are equal to one another, and the…
I.35Proposition I.35Parallelograms which are on the same base and in the same parallels are equal to one…
I.36Proposition I.36Parallelograms which are on equal bases and in the same parallels are equal to one…
I.37Proposition I.37Triangles which are on the same base and in the same parallels are equal to one another.
I.38Proposition I.38Triangles which are on equal bases and in the same parallels are equal to one another.
I.39Proposition I.39Equal triangles which are on the same base and on the same side are also in the same…
I.40Proposition I.40Equal triangles which are on equal bases and on the same side are also in the same…
I.41Proposition I.41If a parallelogram have the same base with a triangle and be in the same parallels, the…
I.42Proposition I.42To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.
I.43Proposition I.43In any parallelogram the complements of the parallelograms about the diameter are equal…
I.44Proposition I.44To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to…
I.45Proposition I.45To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal…
I.46Proposition I.46On a given straight line to describe a square.
I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to…
I.48Proposition I.48If in a triangle the square on one of the sides be equal to the squares on the remaining…
II.1Proposition II.1If there be two straight lines, and one of them be cut into any number of segments…
II.2Proposition II.2If a straight line be cut at random, the rectangle contained by the whole and both of the…
II.3Proposition II.3If a straight line be cut at random, the rectangle contained by the whole and one of the…
II.4Proposition II.4If a straight line be cut at random, the square on the whole is equal to the squares on…
II.5Proposition II.5If a straight line be cut into equal and unequal segments, the rectangle contained by the…
II.6Proposition II.6If a straight line be bisected and a straight line be added to it in a straight line, the…
II.7Proposition II.7If a straight line be cut at random, the square on the whole and that on one of the…
II.8Proposition II.8If a straight line be cut at random, four times the rectangle contained by the whole and…
II.9Proposition II.9If a straight line be cut into equal and unequal segments, the squares on the unequal…
II.10Proposition II.10If a straight line be bisected and a straight line be added to it in a straight line, the…
II.11Proposition II.11To cut a given straight line so that the rectangle contained by the whole and one of the…
II.12Proposition II.12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater…
II.13Proposition II.13In acute-angled triangles the square on the side subtending the acute angle is less than…
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.2Proposition III.2If on the circumference of a circle two points be taken at random, the straight line…
III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the…
III.4Proposition III.4If in a circle two straight lines cut one another which are not through the centre, they…
III.5Proposition III.5If two circles cut one another, they will not have the same centre.
III.6Proposition III.6If two circles touch one another, they will not have the same centre.
III.7Proposition III.7If on the diameter of a circle a point be taken which is not the centre, and from the…
III.8Proposition III.8If a point be taken outside a circle and from the point straight lines be drawn through…
III.9Proposition III.9If a point be taken within a circle, and more than two equal straight lines fall from the…
III.10Proposition III.10A circle does not cut a circle at more points than two.
III.11Proposition III.11If two circles touch one another internally, and their centres be taken, the straight…
III.12Proposition III.12If two circles touch one another externally, the straight line joining their centres will…
III.13Proposition III.13A circle does not touch a circle at more points than one, whether it touch it internally…
III.14Proposition III.14In a circle equal straight lines are equally distant from the centre, and those which are…
III.15Proposition III.15Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the…
III.16Proposition III.16The straight line drawn at right angles to the diameter of a circle from its extremity…
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…
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