Elements of geometry, containing books i. to vi.and portions of books xi. and xii. of Euclid, with exercises and notes, by J.H. Smith |
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Página 7
... shew hereafter , be deduced from a more simple Postulate . The student must defer the consideration of this Postulate , till he has reached the 17th Proposition of Book I. Euclid next enumerates , as statements of fact , nine Axioms or ...
... shew hereafter , be deduced from a more simple Postulate . The student must defer the consideration of this Postulate , till he has reached the 17th Proposition of Book I. Euclid next enumerates , as statements of fact , nine Axioms or ...
Página 12
... Shew that if straight lines be drawn from A and B in the diagram of Prop . I. to the other point in which the circles intersect , another equilateral triangle will be described on AB . 2. By a construction similar to that in Prop . III ...
... Shew that if straight lines be drawn from A and B in the diagram of Prop . I. to the other point in which the circles intersect , another equilateral triangle will be described on AB . 2. By a construction similar to that in Prop . III ...
Página 20
... Shew that we can prove this Proposition by means of Prop . IV . and PROP . A. , without applying Prop . C. Ex . 2. If the equilateral triangle , employed in the construc- tion , be described with its vertex towards the given angle ; shew ...
... Shew that we can prove this Proposition by means of Prop . IV . and PROP . A. , without applying Prop . C. Ex . 2. If the equilateral triangle , employed in the construc- tion , be described with its vertex towards the given angle ; shew ...
Página 22
... Shew that in the diagram of Prop . ix . AF and ED intersect each other at right angles , and that ED is bisected by ... shew that OA , OB , OC are all equal . Ex . 3. Shew that Prop . XI . is a particular case of Prop . IX . PROPOSITION ...
... Shew that in the diagram of Prop . ix . AF and ED intersect each other at right angles , and that ED is bisected by ... shew that OA , OB , OC are all equal . Ex . 3. Shew that Prop . XI . is a particular case of Prop . IX . PROPOSITION ...
Página 24
... shew that BDC is an isosceles triangle . 4. D , E , F are points taken in the sides BC , CA , AB , of an equilateral triangle , so that BD - CE = AF . Shew that the triangle DEF is equilateral . 5. In a given straight line find a point ...
... shew that BDC is an isosceles triangle . 4. D , E , F are points taken in the sides BC , CA , AB , of an equilateral triangle , so that BD - CE = AF . Shew that the triangle DEF is equilateral . 5. In a given straight line find a point ...
Términos y frases comunes
AB=DE ABCD AC=DF angles equal angular points base BC BC=EF centre chord circumference coincide described diagonals diameter divided draw equal angles equiangular equilateral triangle equimultiples Eucl Euclid exterior angle given circle given line given point given straight line greater than nB Hence hypotenuse inscribed intersect isosceles triangle less Let ABC Let the st lines be drawn magnitudes middle points multiple opposite angles opposite sides parallel parallelogram pentagon perpendicular plane polygon produced Prop prove Q. E. D. Ex Q. E. D. PROPOSITION quadrilateral radius ratio rectangle contained reflex angle rhombus right angles segment shew shewn square subtended sum of sqq tangent THEOREM together=two rt trapezium triangle ABC triangles are equal vertex vertical angle
Pasajes populares
Página 89 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Página 168 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.
Página 7 - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Página 40 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Página 23 - To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.
Página 106 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line ; it is required to draw a straight line through the point A, parallel to the straight hue BC.
Página 178 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Página 46 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another...
Página 285 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Página 91 - In every triangle, the square of the side subtending either of the acute angles is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.