Posts tagged 'rectangle'

G.81 contained rectangle

Any rectangular parallelogram is said to be:term:**contained** <contained-rectangle> by the two:term:segments <G.4> containing the:term:right angle <G.33>.

G.82 distributive law of rectangles

If one of two segments is cut into any number of parts, the rectangle contained by the two segments equals the sum of the rectangles contained by the uncut segment and each of the parts.

G.83 square equals sum of rectangles on parts

If a segment is cut at random, the sum of the rectangles contained by the whole and each of the parts equals the square on the whole.

G.84 rectangle on whole and part

If a segment is cut at random, the rectangle contained by the whole and one of the parts equals the rectangle contained by the parts together with the square on that part.

G.85 square of a sum

If a segment is cut at random, the square on the whole equals the squares on the parts together with twice the rectangle contained by the parts.

G.87 rectangle on unequal parts plus square on difference

If a segment is cut into equal and unequal parts, the rectangle contained by the unequal parts together with the square on the segment between the points of section equals the square on the half.

G.88 rectangle on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the rectangle contained by the whole with the added segment and the added segment together with the square on the half equals the square on the segment made up of the half and the added segment.

G.89 square of a difference

If a segment is cut at random, the sum of the squares on the whole and on one of the parts equals twice the rectangle contained by the whole and that part together with the square on the remaining part.

G.90 four rectangles plus square on difference

If a segment is cut at random, four times the rectangle contained by the whole and one of the parts together with the square on the remaining part equals the square on the whole and the first part as one segment.

G.93 cut a segment in extreme and mean ratio

Cut a given segment so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

G.94 law of cosines for obtuse triangles

In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.95 law of cosines for acute triangles

In acute-angled triangles, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.143 intersecting chords theorem

If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.

G.144 tangent-secant theorem

If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.

G.145 converse of the tangent-secant theorem

If from a point outside a circle a secant and a segment to the circle are drawn such that the rectangle on the whole secant and its external part equals the square on the segment, the segment is tangent to the circle.

G.227 segments proportional rectangle

If four segments be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, the four segments is proportional.

G.228 segments proportional rectangle

If three segments be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, the three segments is proportional.

G.393 rectangle contained rational

The rectangle contained by rational segments commensurable in length is rational.

G.395 rectangle contained rational

The rectangle contained by rational segments commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.

G.398 rectangle contained medial

The rectangle contained by medial segments commensurable in length is medial.

G.399 rectangle contained medial

The rectangle contained by medial segments commensurable in square only is either rational or medial.

G.401 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a rational rectangle.

G.402 find medial segments commensurable in square only which

Find medial segments commensurable in square only which contain a medial rectangle.

G.405 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a rational rectangle, and such that the square on the greater is greater than the square on the less by the square on a segment commensurable in length with the greater.

G.406 find two medial segments commensurable in square only

Find two medial segments commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater.

G.407 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.

G.408 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

G.409 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.

G.411 medial segments commensurable

If two medial segments commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial segment.

G.412 medial segments commensurable

If two medial segments commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial segment.

G.413 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole segment is irrational: and let it be called major.

G.414 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole segment is irrational; and let it be called the side of a rational plus a medial area.

G.415 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole segment is irrational; and let it be called the side of the sum of two medial areas.

G.455 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial segment.

G.457 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial segment.

G.458 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called minor.

G.459 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational: and let it be called that which produces with a rational area a medial whole.

G.460 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

G.462 apotome medial segment

To a first apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.

G.463 apotome medial segment

To a second apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle.

G.464 minor segment only

To a minor segment only one segment can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of the squares on them rational but twice the rectangle contained by them medial.

G.465 segment produces rational

To a segment which produces with a rational area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial, but twice the rectangle contained by them rational.

G.562 cylinder

When, one side of those about the:term:right angle <G.33> in a:term:rectangular parallelogram <G.81> remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the:term:figure <G.9> so comprehended is a:term:**cylinder** <cylinder>.