A companion to Euclid: being a help to the understanding and remembering of the first four books. With a set of improved figures, and an original demonstration of the proposition called in Euclid the twelfth axiom, by a graduateJohn W. Parker, 1837 - 88 Seiten |
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Seite 55
... point K within △ MLD ) are the side of the A , ML + LD , 3. that DK < DL , 4. that similarly DL < DH . B * If from a point outside a circle , as A , two lines AB , AC be drawn touching the circle , and the points of contact be joined ...
... point K within △ MLD ) are the side of the A , ML + LD , 3. that DK < DL , 4. that similarly DL < DH . B * If from a point outside a circle , as A , two lines AB , AC be drawn touching the circle , and the points of contact be joined ...
Seite 57
... point of contact . Steps of the Demonstration . Suppose that the line which joins the centres does not pass through the point of contact A , but has some other direction , as BC ; then prove , on that supposition , 1. That FG + GA > FB ...
... point of contact . Steps of the Demonstration . Suppose that the line which joins the centres does not pass through the point of contact A , but has some other direction , as BC ; then prove , on that supposition , 1. That FG + GA > FB ...
Seite 58
... point of contact . E D Prove that the supposition , that the line joining the centres passes otherwise than through ... points than one , whether it touches on the inside or the outside . H B E E H B F F Steps of the Demonstration to ...
... point of contact . E D Prove that the supposition , that the line joining the centres passes otherwise than through ... points than one , whether it touches on the inside or the outside . H B E E H B F F Steps of the Demonstration to ...
Seite 59
... point , as в and » , show , on that supposition , 1. That right line BD falls within each O , 2. that GH passes through the point of contact , which shows the supposition to be false . K B Steps of the Demonstration to Case 2nd . Having ...
... point , as в and » , show , on that supposition , 1. That right line BD falls within each O , 2. that GH passes through the point of contact , which shows the supposition to be false . K B Steps of the Demonstration to Case 2nd . Having ...
Seite 63
... point D is on the of the O , is proved by Cor . to xvi . 3 . PROPOSITION XVIII . ( Argument ad absurdum ) . B D G I Theorem . If a straight line touches a circle , the straight line drawn from the centre to the point of contact shall be ...
... point D is on the of the O , is proved by Cor . to xvi . 3 . PROPOSITION XVIII . ( Argument ad absurdum ) . B D G I Theorem . If a straight line touches a circle , the straight line drawn from the centre to the point of contact shall be ...
Häufige Begriffe und Wortgruppen
AB² AC² AD² AEX EC angle contained angle equal Argument ad absurdum base DF BC² BD² bisect CB² cuts the circle DC² Demonstration itself consists diameter EB² EF² EG² Engravings equal straight lines equi equiangular equilateral Euclid F Steps fall figure GF² given circle given point given rectilineal angle given straight line given triangle i. e. less inscribe interior angles learner less greater line be divided line drawn parallel parallelogram PARKER pass pentagon point of contact Problem proof PROPOSITION IX PROPOSITION VIII Proved by showing rectangle contained right angles right line shows the supposition similarly Suppose supposition is false Theorem WEST STRAND whole line
Beliebte Passagen
Seite 24 - 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, namely, either t}le sides adjacent to the equal...
Seite 45 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part.
Seite 18 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.
Seite 61 - From this it is manifest that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle...
Seite 37 - 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.
Seite 76 - 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.
Seite 77 - If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on GEOMETRY.
Seite 72 - If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.
Seite 27 - If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.