Proof
Erect squares on each side via I.46; using I.14, I.31 and I.41 show
each part-square equals a corresponding parallelogram cut off the
hypotenuse-square by the perpendicular from the right angle; sum via
Common Notion 2.
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Full neighborhood
Depends on (6)
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
- I.14Proposition I.14If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles…
- I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
- I.41Proposition I.41If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the…
- I.46Proposition I.46On a given straight line to describe a square.
- 2Common notion 2If equals be added to equals, the wholes are equal.
Required by (dependents) (14)
- 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.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.12Proposition II.12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides…
- II.13Proposition II.13In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides…
- II.14Proposition II.14To construct a square equal to a given rectilineal figure.
- III.14Proposition III.14In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the…
- III.15Proposition III.15Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than…
- III.35Proposition III.35If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the…
- III.36Proposition III.36If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut…
- 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…
- XIII.10Proposition XIII.10If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on…
- XIII.12Proposition XIII.12If an equilateral triangle be inscribed in a circle, the square on the side of the triangle is triple of the square on…
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