The First Six Books of the Elements of Euclid: And Propositions I-XXI of Book XI, and an Appendix on the Cylinder, Sphere, Cone, Etc., with Copious Annotations and Numerous ExercisesHodges, Figgis, & Company, 1885 - 312 páginas |
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Página 25
... Prove this Proposition without using Prop . VIII . 2. Prove that AF is perpendicular to DE . 3. Prove that any point in AF is equally distant from the points D and E. 4. Prove that any point in AF is equally distant from the lines AB ...
... Prove this Proposition without using Prop . VIII . 2. Prove that AF is perpendicular to DE . 3. Prove that any point in AF is equally distant from the points D and E. 4. Prove that any point in AF is equally distant from the lines AB ...
Página 27
... Prove Prop . XI . without using Prop . viii . 3. Erect a line at right angles to a given line at one of its ex- tremities without producing the line . 4. Find a point in a given line that shall be equally distant from two given points ...
... Prove Prop . XI . without using Prop . viii . 3. Erect a line at right angles to a given line at one of its ex- tremities without producing the line . 4. Find a point in a given line that shall be equally distant from two given points ...
Página 28
... Prove that the circle cannot meet AB in more than two points . 2. If one angle of a triangle be equal to the sum of the other two , the triangle can be divided into the sum of two isosceles triangles , and the base is equal to twice the ...
... Prove that the circle cannot meet AB in more than two points . 2. If one angle of a triangle be equal to the sum of the other two , the triangle can be divided into the sum of two isosceles triangles , and the base is equal to twice the ...
Página 31
... Prove it . 8. What is the subject of Props . XIII . , XIV . , XV . ? Ans . Angles at a point . PROP . XVI . - THEOREM . If any side ( BC ) of a triangle ( ABC ) be produced , the exterior angle ( ACD ) is greater than either of the inte ...
... Prove it . 8. What is the subject of Props . XIII . , XIV . , XV . ? Ans . Angles at a point . PROP . XVI . - THEOREM . If any side ( BC ) of a triangle ( ABC ) be produced , the exterior angle ( ACD ) is greater than either of the inte ...
Página 32
... acute angles . Cor . 2. - If two angles of a triangle be unequal , the lesser must be acute . Exercise . Prove Prop . XVII . without producing a side . PROP . XVIII . — THEOREM . If in any 32 [ BOOK I. THE ELEMENTS OF EUCLID .
... acute angles . Cor . 2. - If two angles of a triangle be unequal , the lesser must be acute . Exercise . Prove Prop . XVII . without producing a side . PROP . XVIII . — THEOREM . If in any 32 [ BOOK I. THE ELEMENTS OF EUCLID .
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Términos y frases comunes
ABCD AC is equal 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 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
Pasajes populares
Página 299 - Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc.
Página 186 - 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...
Página 9 - LET it be granted that a straight line may be drawn from any one point to any other point.
Página 104 - 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.
Página 124 - 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...
Página 230 - 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.
Página 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.
Página 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.
Página 128 - 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.
Página 22 - ACB, DB is equal to AC, and BC common to both ; the two sides DB, BC, are equal to the two AC, CB, each to each, and the angle DBC is equal to the angle ACB : therefore, the base DC is equal to the base AB, and the triangle DBC (Mr. Southey) is equal to the triangle ACB, the less to the greater, which is absurd,