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**Constructing a Tangent to a Circle

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using a straight edge and a compass

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Step 1: Construct a circle with a radius of any size. Name the
circle C1.

Step 2: Construct a point on the circle. Call it D.

Step 3: Construct a point off of the circle. Call it E.
See sketch 1.

Step 4: Construct a line through point D and point E. Name the
line m.

Step 5: Construct a line through the center of the circle and point
E.
Name the line n.
See sketch 2.

Step 6: (Hide the lines.) Construct a line segment between point
P and
the center of the circle C1.

Step 7: Construct the midpoint of the segment. Name the
midpoint J.
See sketch 3.

Step 8: Construct a circle using the center and a point. The
center will be at
point J while the radius is the distance between point J
and the center of
the circle C1. (Select J first and then while holding the
shift also select
the center of the circle C1.)
See sketch 4.

Step 9: Construct the intersection between the two circles. Name
these points L
and K.
See sketch 5.

Step 10: (Hide the circle whose center is J and hide the line
segment.)
Construct a line through L and P.
(For the other tangent, construct a line through K and P.)
See sketch 6.

Finally, let's prove that these line are tangents to C1.
Remember that the tangent line(s) is(are) perpendicular to radii.
Construct a line segment between L and the center of the circle C1.
Now measure angle L. See sketch 7. The measurement of a
perpendicular angle is 90¡.

Click on the next window to see the actual construction of two
tangents to a circle!
Remeber to start with three points. The first point is the center of
your circle. The second point is a point on the circle. The last point
is a point outside your circle. (Hold the shift key while you are
creating your points.) Start the script.