A School Euclid. Being Books I.&II. of Euclid's Elements. With Notes, Exercises and Explanations ... By C. Mansford1874 |
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Page 21
... joining A with C instead of with B. Draw the figure of this proposition when the given point is in the given line produced . PROPOSITION 3. PROBLEM . From the greater of two given straight lines , to cut off a part equal to the less . 1 ...
... joining A with C instead of with B. Draw the figure of this proposition when the given point is in the given line produced . PROPOSITION 3. PROBLEM . From the greater of two given straight lines , to cut off a part equal to the less . 1 ...
Page 24
... join FC , GB . 3 . Because A Fis equal to AG , [ Con . ] and AB to AC , [ Hyp . ] the two sides FA , AC are equal to the two sides GA , AB , each to each ; and they contain the angle FAG common to the two tri- angles AFC , AGB ; there ...
... join FC , GB . 3 . Because A Fis equal to AG , [ Con . ] and AB to AC , [ Hyp . ] the two sides FA , AC are equal to the two sides GA , AB , each to each ; and they contain the angle FAG common to the two tri- angles AFC , AGB ; there ...
Page 25
... the less , [ I. 3. ] and join DC . B A Then , because in the triangles DBC , ACB , DB is equal to AC , [ Con . ] and BC is common to both , the two sides DB , BC are equal to the two sides 3 . AC , CB , each to each ; and the BOOK I. 6 .
... the less , [ I. 3. ] and join DC . B A Then , because in the triangles DBC , ACB , DB is equal to AC , [ Con . ] and BC is common to both , the two sides DB , BC are equal to the two sides 3 . AC , CB , each to each ; and the BOOK I. 6 .
Page 27
... Join CD . Then in the case in which the vertex of each triangle is without the other triangle ; because AC is equal to AD [ Hyp . ] , the angle ACD is equal to the angle ADC . [ I. 5. ] But the angle ACD is greater than the angle BCD ...
... Join CD . Then in the case in which the vertex of each triangle is without the other triangle ; because AC is equal to AD [ Hyp . ] , the angle ACD is equal to the angle ADC . [ I. 5. ] But the angle ACD is greater than the angle BCD ...
Page 29
... Join AG . Then the angle BAG = BGA because BA = BG , and the angle GAC AGC because AC - GC . Therefore the whole angle BAC BGC , that is EDF . Q.E.D. Ex . 1. The diagonal of a rhombus bisects each of the angles through which it passes ...
... Join AG . Then the angle BAG = BGA because BA = BG , and the angle GAC AGC because AC - GC . Therefore the whole angle BAC BGC , that is EDF . Q.E.D. Ex . 1. The diagonal of a rhombus bisects each of the angles through which it passes ...
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A School Euclid, Being Books I. & II. of Euclid's Elements, with Notes by C ... Euclides No preview available - 2015 |
Common terms and phrases
ABCD AC is equal angle ABC angle ACB angle BAC angle EDF angle equal apply axioms base BC bisect BOOK called centre circle coincide common Const construction definitions demonstration describe diagonals diameter difference divided double draw drawn equal sides equal to AC equilateral triangle Euclid exercise exterior angle fall figure fore geometry given point given rectilineal given straight line gnomon greater Hence isosceles triangle join length less Let ABC meet method namely opposite angle opposite sides parallel parallel to CD parallelogram perpendicular PROBLEM produced prop PROPOSITION proved quadrilateral reason rectangle contained rectilineal figure remainder result right angles side BC sides square on AC Take THEOREM things triangle ABC true truths twice the rectangle unequal units whole
Popular passages
Page 90 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part.
Page 28 - If two triangles have two sides of the one equal to two sides of the...
Page 41 - Any two sides of a triangle are together greater than the third side.
Page 57 - 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.
Page 72 - To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. Let ABCD be the given rectilineal figure, and E the given rectilineal angle. It is required to describe a parallelogram equal to ABCD, and having an angle equal to E.
Page 75 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
Page 89 - 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...
Page 55 - 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 28 - A two triangles, having the two sides AB, AC, equal to the two sides \ DE, DF, each to each, viz: AB to DE, and AC to DF; and also the base BC equal to the base EF.
Page 69 - 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.