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algebra ancient angle appears applications attempt axioms base beginning better Book called century CHAPTER circle common considered construction course define definition distance drawing drawn easily educational effort elementary equal Euclid example exercises fact figure four give given greater Greek idea illustration important included inscribed interest intersect kind known later lead less limit mathematics means measure method mind nature necessary needed parallel perpendicular plane geometry polygon possible postulate practical present probably problem Proclus produced proof proportion proposition proved pupil question ratio reason regular relating result right angles seems seen shown shows sides similar simple solid geometry space speak sphere square statement straight line suggested surface taken tangent teacher teaching textbook THEOREM thing third tion triangle usually volume writers
Side 254 - 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.
Side 125 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Side 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.
Side 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 greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
Side 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.
Side 244 - If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Side 225 - Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Side 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.