Posts tagged 'porism'

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G.118 perpendicular at extremity of a diameter is tangent

The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.
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G.232 similar polygons divided

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.
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G.381 squares segments commensurable

The squares on segments commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on segments incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
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G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.
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G.577 spheres center inscribe

Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.
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G.597 construct an icosahedron and comprehend it in a

Construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational segment called minor.
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G.600 construct a dodecahedron and comprehend it in a

Construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational segment called apotome.