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algebra altitude ancient angles are equal applications Archimedes Aristotle attempt axioms basal base beginning better bisect Book called century A.D. CHAPTER circle circumference congruent considered construction corollary cube curve cylinder define definition distance drawing easily edge elementary geometry equilateral etry Euclid Euclid's Elements example exercises fact figure geom given line Greek Heron of Alexandria hexagon illustration incommensurable inscribed interest intersect isosceles known locus logic mathematician mathematics means measure method modern octahedron parallel parallelepiped parallelogram perpendicular plane geometry Plato polyhedrons postulate practical prism problem Proclus proof proportion proposition proved pupil pyramid Pythagoras Pythagorean Theorem question radius ratio reason regular polygons relating right angles right triangle segment sides sition solid geometry sphere spherical square statement straight line surface syllabus tangent teacher teaching textbook Thales thing tion to-day trigonometry usually vertex vertices volume word writers
Page 256 - Two triangles having an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.
Page 143 - A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; 16 And the point is called the center of the circle.
Page 145 - ... 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.
Page 246 - If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Page 229 - Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Page 188 - If two triangles have two sides of the one equal, respectively, to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
Page 142 - But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it (Def.
Page 54 - Euclid's, and show by construction that its truth was known to us ; to demonstrate, for example, that the angles at the base of an isosceles triangle are equal...
Page 299 - If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal, and their planes are parallel. Let the corresponding sides of angles A and A' in the planes MN and PQ be parallel, and lie on the same side of AA'.