The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and ExercisesMacmillan, 1867 - 400 páginas |
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Página 277
... intersect at O , and the rectangle AO , OB be equal to the rectangle CO , OD , the circum- ference of a circle will pass through the four points A , B , C , D. For a circle may be described round the triangle ABC , by IV . 5 ; and then ...
... intersect at O , and the rectangle AO , OB be equal to the rectangle CO , OD , the circum- ference of a circle will pass through the four points A , B , C , D. For a circle may be described round the triangle ABC , by IV . 5 ; and then ...
Página 293
... , DB , and twice the square on DC , together with twice the rectangle AD , DE . But AD is equal to DB . Therefore the squares on AC , BC are equal to twice the squares on AD , DC . 2. If two chords intersect within a circle , the.
... , DB , and twice the square on DC , together with twice the rectangle AD , DE . But AD is equal to DB . Therefore the squares on AC , BC are equal to twice the squares on AD , DC . 2. If two chords intersect within a circle , the.
Página 294
... intersect within a circle , the angle which they include is measured by half the sum of the in- tercepted arcs . Let the chords AB and CD of a circle intersect at E ; join AD . The angle AEC is equal to the angles ADE , and DAE , by I ...
... intersect within a circle , the angle which they include is measured by half the sum of the in- tercepted arcs . Let the chords AB and CD of a circle intersect at E ; join AD . The angle AEC is equal to the angles ADE , and DAE , by I ...
Página 295
... intersect . When each of the given circles is without the other we can obtain two other solutions . For , describe a circle with A as a centre and radius equal to the sum of the radii of the given circles ; and continue as before ...
... intersect . When each of the given circles is without the other we can obtain two other solutions . For , describe a circle with A as a centre and radius equal to the sum of the radii of the given circles ; and continue as before ...
Página 305
... intersect the smaller circle again at K ; then AK is parallel to BH ( 14 ) ; therefore the angle AKT is equal to the angle BHG ; and the angle AKG is equal to the angle AGK , which is equal to the angle OGH , which is equal to the angle ...
... intersect the smaller circle again at K ; then AK is parallel to BH ( 14 ) ; therefore the angle AKT is equal to the angle BHG ; and the angle AKG is equal to the angle AGK , which is equal to the angle OGH , which is equal to the angle ...
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Términos y frases comunes
ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral radius rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Pasajes populares
Página 286 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Página 30 - 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 34 - 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...
Página 37 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Página 12 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Página 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Página 220 - 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 354 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.
Página 104 - 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 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.