The first six books of the Elements of Euclid, and propositions i.-xxi. of book xi1885 |
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Σελίδα 18
... , taken along with iv . 8. What property of two lines having two common points is quoted in this Proposition ? They must coincide . Exercises . 1. The line that bisects the vertical angle 18 [ BOOK I , THE ELEMENTS OF EUCLID .
... , taken along with iv . 8. What property of two lines having two common points is quoted in this Proposition ? They must coincide . Exercises . 1. The line that bisects the vertical angle 18 [ BOOK I , THE ELEMENTS OF EUCLID .
Σελίδα 19
... bisects the base perpendicularly . 2. If two adjacent sides of a quadrilateral be equal , and the diagonal bisects the angle between them , their other sides are equal . 3. If two lines be at right angles , and if each bisect the other ...
... bisects the base perpendicularly . 2. If two adjacent sides of a quadrilateral be equal , and the diagonal bisects the angle between them , their other sides are equal . 3. If two lines be at right angles , and if each bisect the other ...
Σελίδα 24
... because A D B E BGBA , the angle BAG BGA . In like manner the angle CAG - CGA . Hence the whole angle BAC BGC ; but BGC EDF . therefore BAC = EDF . - = PROP . IX . - PROBLEM . To bisect a 24 [ воок г. THE ELEMENTS OF EUCLID .
... because A D B E BGBA , the angle BAG BGA . In like manner the angle CAG - CGA . Hence the whole angle BAC BGC ; but BGC EDF . therefore BAC = EDF . - = PROP . IX . - PROBLEM . To bisect a 24 [ воок г. THE ELEMENTS OF EUCLID .
Σελίδα 25
... bisects the given angle BAC . Dem . The triangles DAF , EAF have the side AD equal to AE ( const . ) and AF common ... bisect a given BOOK I. ] THE ELEMENTS OF EUCLID . 25.
... bisects the given angle BAC . Dem . The triangles DAF , EAF have the side AD equal to AE ( const . ) and AF common ... bisect a given BOOK I. ] THE ELEMENTS OF EUCLID . 25.
Σελίδα 26
Euclides John Casey. PROP . X. - PROBLEM . To bisect a given finite right line ( AB ) . Sol . - Upon AB describe an equilateral triangle ACB [ 1 ] Bisect the angle ACB by the line CD [ Ix . ] , meeting AB in D , then AB is bisected in D ...
Euclides John Casey. PROP . X. - PROBLEM . To bisect a given finite right line ( AB ) . Sol . - Upon AB describe an equilateral triangle ACB [ 1 ] Bisect the angle ACB by the line CD [ Ix . ] , meeting AB in D , then AB is bisected in D ...
Συχνά εμφανιζόμενοι όροι και φράσεις
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
Δημοφιλή αποσπάσματα
Σελίδα 295 - Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc.
Σελίδα 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...
Σελίδα 9 - LET it be granted that a straight line may be drawn from any one point to any other point.
Σελίδα 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.
Σελίδα 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...
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.