The first six books of the Elements of Euclid, and propositions i.-xxi. of book xi1885 |
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Page 14
... triangle . Sol . - With A as centre , and AB as radius , describe the circle BCD ( Post . I. ) . With B as centre , and BA as radius , describe the circle ACE , cutting the former circle in C. Join CA , CB ( Post . I. ) . Then ABC is ...
... triangle . Sol . - With A as centre , and AB as radius , describe the circle BCD ( Post . I. ) . With B as centre , and BA as radius , describe the circle ACE , cutting the former circle in C. Join CA , CB ( Post . I. ) . Then ABC is ...
Page 19
... triangle bisects the base perpendicularly . 2. If two adjacent sides of a quadrilateral be equal , and the diagonal ... ( ABC , ACB ) at the base ( BC ) of an isosceles triangle are equal to one another , and if the equal sides ( AB , AC ) ...
... triangle bisects the base perpendicularly . 2. If two adjacent sides of a quadrilateral be equal , and the diagonal ... ( ABC , ACB ) at the base ( BC ) of an isosceles triangle are equal to one another , and if the equal sides ( AB , AC ) ...
Page 20
... ABC , and these are the angles at the base . A A Observation . The great difficulty which beginners find in this ... triangle is evidently equal to the angle ABC , with which it originally coincided . Again , the two As BAC , CAD have ...
... ABC , and these are the angles at the base . A A Observation . The great difficulty which beginners find in this ... triangle is evidently equal to the angle ABC , with which it originally coincided . Again , the two As BAC , CAD have ...
Page 21
... ABC ; therefore ACB is equal to ABC . Cor . - Every equilateral triangle is equiangular . DEF . - A line in any figure , such as AC in the preceding diagram , which is such that , by folding the plane of the figure round it , one part ...
... ABC ; therefore ACB is equal to ABC . Cor . - Every equilateral triangle is equiangular . DEF . - A line in any figure , such as AC in the preceding diagram , which is such that , by folding the plane of the figure round it , one part ...
Page 23
... triangle ACD is isosceles , and [ v . ] the external angles ECD , FDC at the ... ( ABC , DEF ) have two sides ( AB , AC ) of one respectively equal to two ... triangle ABC be applied to DEF , so that the point B will coincide with E , and ...
... triangle ACD is isosceles , and [ v . ] the external angles ECD , FDC at the ... ( ABC , DEF ) have two sides ( AB , AC ) of one respectively equal to two ... triangle ABC be applied to DEF , so that the point B will coincide with E , and ...
Common terms and phrases
ABCD AC is equal AD² adjacent angles altitude angle ABC angle ACB angle BAC angular points Axiom bisector bisects centre chord circles touch circumference circumscribed circle collinear concurrent lines const coplanar cyclic quadrilateral Dem.-Let diagonals diameter divided draw equal angles equal to AC equiangular equilateral triangle escribed circles Euclid Exercises exterior angle Geometry given circle given line given point greater Hence the angle hypotenuse inscribed isosceles less line AC line joining locus manner meet middle points multiple nine-points circle opposite sides parallel parallelogram parallelopiped perpendicular plane points of intersection prism PROP Proposition prove radii radius rectangle contained rectilineal figure regular polygon respectively equal right angles right line segments semicircle sides AC similar square on AC tangent theorem triangle ABC vertex vertical angle
Popular passages
Page 295 - Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc.
Page 182 - When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth...
Page 9 - LET it be granted that a straight line may be drawn from any one point to any other point.
Page 102 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square on the other part.
Page 122 - The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less. Let ABCD be a circle, of which...
Page 226 - If from any angle of a triangle, a straight line be drawn perpendicular to the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.
Page 29 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.
Page 63 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 126 - The diagonals of a quadrilateral intersect at right angles. Prove that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair.
Page 194 - If there be any number of proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.