Elements of geometry, based on Euclid, books i-iii1876 - 119 páginas |
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Página 105
... chords cut off equal arcs , the greater equal to the greater , and the less equal to the less . Let ABC , DEF be equal circles , and BC , EF equal chords in them , which cut off the two greater arcs BAC , EDF , and the two less arcs BGC ...
... chords cut off equal arcs , the greater equal to the greater , and the less equal to the less . Let ABC , DEF be equal circles , and BC , EF equal chords in them , which cut off the two greater arcs BAC , EDF , and the two less arcs BGC ...
Página 106
... chords . Let ABC , DEF be equal circles , and let BGC , EHF be equal arcs in them , and join BC , EF . The chord BC shall be equal to the chord EF . CONSTRUCTION . - Take K , L , the centres of the circles and L the ( III . 1 ) , and ...
... chords . Let ABC , DEF be equal circles , and let BGC , EHF be equal arcs in them , and join BC , EF . The chord BC shall be equal to the chord EF . CONSTRUCTION . - Take K , L , the centres of the circles and L the ( III . 1 ) , and ...
Página 107
... chords cut off equal arcs , the greater equal to the greater , and the less equal to the less ( III . 28 ) ; And each of the arcs AD , DB is less than a semicircle , because DC , if produced , is a diameter ( III . 1 , cor ...
... chords cut off equal arcs , the greater equal to the greater , and the less equal to the less ( III . 28 ) ; And each of the arcs AD , DB is less than a semicircle , because DC , if produced , is a diameter ( III . 1 , cor ...
Página 108
... chord drawn from the point of contact are equal to the angles in the alternate segments of the circle . Let EF be a tangent to the circle ABCD , and BD a chord drawn from the point of contact B , cutting the circle . The angles which BD ...
... chord drawn from the point of contact are equal to the angles in the alternate segments of the circle . Let EF be a tangent to the circle ABCD , and BD a chord drawn from the point of contact B , cutting the circle . The angles which BD ...
Página 118
... chord through the point of contact is equal to half the angle which the chord sub- tends at the centre . 28. From a given point P , within or without a circle , draw a straight line cutting the circle in A and B such that PA shall be ...
... chord through the point of contact is equal to half the angle which the chord sub- tends at the centre . 28. From a given point P , within or without a circle , draw a straight line cutting the circle in A and B such that PA shall be ...
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Términos y frases comunes
AB is equal AC and CD adjacent angles alternate angles angle ABC angle ACB angle BAC angle BCD angle contained angle DEF angle EDF angle equal angles BGH angles CBA base BC BC is equal bisect centre circle ABC circumference diagonal diameter double draw equal circles equal to AC equal to twice EUCLID'S ELEMENTS exterior angle given circle given point given rectilineal angle given straight line gnomon greater interior and opposite isosceles triangle less Let ABC Let the straight opposite angles parallel to BC parallelogram perpendicular produced PROOF PROOF.-Because Q.E.D. Proposition rectangle AD rectangle AE rectangle contained remaining angle right angle Const right angles Ax segment semicircle side BC square described square on AC touches the circle triangle ABC triangle DEF twice the rectangle
Pasajes populares
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 equal to two right angles.
Página 13 - 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. Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC be produced to D and E.
Página 7 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Página 17 - If two triangles have two sides of the one equal to two sides of the...
Página 53 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Página 9 - 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...
Página 71 - 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 9 - Things which are double of the same, are equal to one another. 7. Things which are halves of the same, are equal to one another.
Página 34 - Wherefore, if a straight line, &c. QED PROPOSITION XXVIII. THEOREM. If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Página 69 - If a straight line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line, which is made up of the half and the part produced.