From a given point to draw a straight line touching a given circle
be the given point, and BCD
the given circle; thus it is required to draw from the point A
a straight line touching the circle BCD
For let the centre E
of the circle be taken; [III. 1
] let AE
be joined, and with centre E
and distance EA
let the circle AFG
be described; from D
be drawn at right angles to EA
, and let EF
be joined; I say that AB
has been drawn from the point A
touching the circle BCD
For, since E
is the centre of the circles BCD
, EA is equal to EF, and ED to EB;
therefore the two sides AE
are equal to the two sides FE
: and they contain a common angle, the angle at E
; therefore the base DF is equal to the base AB, and the triangle DEF is equal to the triangle BEA, and the remaining angles to the remaining angles; [I. 4] therefore the angle EDF is equal to the angle EBA.
But the angle EDF
is right; therefore the angle EBA is also right.
is a radius; and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [III. 16, Por.
] therefore AB touches the circle BCD.
Therefore from the given point A
the straight line AB
has been drawn touching the circle BCD