Posts tagged 'triangle'

I.48

If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.

I.47

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

I.44

To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.

I.42

To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.

I.41

If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.

I.40

Equal triangles which are on equal bases and on the same side are also in the same parallels.

I.39

Equal triangles which are on the same base and on the same side are also in the same parallels.

I.38

Triangles which are on equal bases and in the same parallels are equal to one another.

I.37

Triangles which are on the same base and in the same parallels are equal to one another.

XI.39

If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double of the triangle, the prisms will be equal.

I.32

In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

I.26

If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle.

I.25

If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.

I.24

If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base.

I.22

Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one.

I.21

If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.

I.20

In any triangle two sides taken together in any manner are greater than the remaining one.

I.19

In any triangle the greater angle is subtended by the greater side.

I.18

In any triangle the greater side subtends the greater angle.

XI.22

If there be three plane angles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal straight lines, it is possible to construct a triangle out of the straight lines joining the extremities of the equal straight lines.

I.17

In any triangle two angles taken together in any manner are less than two right angles.

I.16

In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.

I.8

If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.

I.6

If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

I.5

In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

VI.32

If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles will be in a straight line.

I.4

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

VI.31

In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

I.1

On a given finite straight line to construct an equilateral triangle.

XI.2

If two straight lines cut one another, they are in one plane, and every triangle is in one plane.

XI.def.27

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

XI.def.26

An octahedron is a solid figure contained by eight equal and equilateral triangles.

VI.20

Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding

side.

VI.19

Similar triangles are to one another in the duplicate ratio of the corresponding sides.

I.def.21

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acuteangled triangle that which has its three angles acute.

I.def.20

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

VI.15

In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.

XI.def.19

The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

XI.def.18

When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone.

IV.10

To construct an isosceles triangle having each of the angles at the base double of the remaining one.

II.13

In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acutc angle.

II.12

In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the

perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.

VI.8

If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.

XIII.12

If an equilateral triangle be inscribed in a circle, the square on the side of the triangle is triple of the square on the radius of the circle.

VI.7

If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal, the sides about which are proportional.

VI.6

If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

IV.3

About a given circle to circumscribe a triangle equiangular with a given triangle.

VI.5

If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

VI.4

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

IV.2

In a given circle to inscribe a triangle equiangular with a given triangle.

VI.3

If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.

VI.2

If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle.

VI.1

Triangles and parallelograms which are under the same height are to one another as their bases.