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ABCD angle ACB angle BAC axiom base bisected Book chord circumference coincide common construction converse definition demonstrated described diagonals diameter difference distance divided double draw equal equilateral triangle Euclid exterior angle extremities fall figure four Geometry given circle given line given point given straight line greater half Hence inscribed interior intersection isosceles triangle join length less Let ABC line drawn magnitude meet opposite angles opposite sides parallel parallelogram pass perpendicular possible problem produced Prop PROPOSITION proved radius reason rectangle contained regular remaining respectively right angles segment semicircle shew shewn sides square on AC tangent term THEOREM third touch touches the circle triangle ABC unequal units vertex vertical angle wherefore whole
Page 118 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
Page 90 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.
Page 30 - ... twice as many right angles as the figure has sides ; therefore all the angles of the figure together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 54 - If two triangles have two sides of the one equal to two sides of the...
Page 5 - LET it be granted that a straight line may be drawn from any one point to any other point.
Page 85 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.
Page 3 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 96 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side...
Page 41 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Page 126 - EF, that is, AF, is greater than BF : Again, because BE is equal to CE, and FE common to the triangles BEF, CEF, the two sides BE, EF are equal to the two CE, EF; but the angle BEF is greater than the angle CEF ; therefore the base BF is greater (24. 1.) than the base FC ; for the same reason, CF is greater than GF. Again, because GF, FE are greater (20.