Elements of Geometry: With Notes |
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Page 5
... double , triple , & c . , of the same , or of equal magnitudes , are equal to each other . 3. Magnitudes which are each one half , one third , & c . , of the same or of equal magnitudes , are equal to each other . 4. If equals be either ...
... double , triple , & c . , of the same , or of equal magnitudes , are equal to each other . 3. Magnitudes which are each one half , one third , & c . , of the same or of equal magnitudes , are equal to each other . 4. If equals be either ...
Page 16
... double of either angle at the base , and if the base be produced , the ex- terior angle will exceed the interior adjacent angle . Cor . 8. From a point without a straight line , only one per- pendicular to that line can be drawn , for ...
... double of either angle at the base , and if the base be produced , the ex- terior angle will exceed the interior adjacent angle . Cor . 8. From a point without a straight line , only one per- pendicular to that line can be drawn , for ...
Page 29
... double the triangle ( Prop . XXVII . Cor . 3. B. I. ) . Cor . 2. Rectangles contained by equal lines are equal . PROPOSITION III . THEOREM . Rhomboids which have the same base and equal alti- tudes are equivalent . Let the rhomboids AC ...
... double the triangle ( Prop . XXVII . Cor . 3. B. I. ) . Cor . 2. Rectangles contained by equal lines are equal . PROPOSITION III . THEOREM . Rhomboids which have the same base and equal alti- tudes are equivalent . Let the rhomboids AC ...
Page 31
... double the rectangle contained by those parts . The square ABCD upon the line AB , is equivalent to the squares upon any two parts AE , EB , into which the line is divided , together with double the rectangle contained by them . Let EG ...
... double the rectangle contained by those parts . The square ABCD upon the line AB , is equivalent to the squares upon any two parts AE , EB , into which the line is divided , together with double the rectangle contained by them . Let EG ...
Page 34
... double the triangle ABC ( Prop . II . Cor . 1. B. II . ) ; hence the squares AG , AD , are together likewise equal to the whole space BGHDC , diminished by three times the triangle ABC ; consequently these two squares are 34 ELEMENTS OF ...
... double the triangle ABC ( Prop . II . Cor . 1. B. II . ) ; hence the squares AG , AD , are together likewise equal to the whole space BGHDC , diminished by three times the triangle ABC ; consequently these two squares are 34 ELEMENTS OF ...
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Common terms and phrases
adjacent angles altitude angle ABC angle ACB angle BAC antecedent base centre chord circ circle circumference circumscribed polygon coincide consequently Prop construction Converse of Prop corollary demonstration described diagonals diameter divided draw equal angles equal Prop equal to AC equimultiples equivalent Euclid exterior angle follows four right angles geometry given straight line gonal greater half hence homologous sides hypothenuse hypothesis included angle inscribed angle inscribed polygon intersect isosceles triangle join Legendre less line drawn lines be drawn magnitudes meet multiple number of sides obtuse opposite angles parallel perimeter perpendicular PROBLEM proportion PROPOSITION XII quadrilateral radii rectangle rectangle contained regular polygon respectively equal rhomboid right angled triangle Scholium side BC similar polygons similar triangles submultiple subtended surface tangent THEOREM three angles tiple triangle ABC vertex VIII
Popular passages
Page 165 - ... if a straight line, &c. QED PROPOSITION 29. — Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles.
Page 172 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Page 30 - 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. Let...
Page 185 - FBC ; and because the two sides AB, BD are equal to the two FB, BC, each to each, and the angle DBA equal to the angle FBC; therefore the base AD is equal (i.
Page 86 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...
Page 142 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Page 205 - Let AMB be the enveloped line; then will it be less than the line APDB which envelopes it. We have already said that by the term convex line we understand a line, polygonal, or curve, or partly curve and . partly polygonal, such that a straight line cannot cut it in more than two points.
Page 185 - BK, it is demonstrated that the parallelogram CL is equal to the square HC. Therefore the whole square BDEC is equal to the two squares GB, HC ; and the square BDEC is described upon the straight line BC, and the squares GB, HC upon BA, AC.
Page 105 - And since a radius drawn to the point of contact is perpendicular to the tangent, it follows that the angle included by two tangents, drawn from the same point, is bisected by a line drawn from the centre of the circle to that point ; for this line forms the hypotenuse common to two equal right angled triangles. PROP. XXXVII. THEOR. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it ; if the rectangle...
Page 35 - In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.