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The method in Euclidean Geometry to construct
two tangents to a circle from a point outside the circle (see
the left-hand part of Figure 19) cannot be applied to the case
under the upper half-plane model (see the right-hand part of Figure
19).
 Figure 19
We will start this construction first and explain
it later.
Given:
A u-circle c and a point A outside c in the
upper half-plane.
To Construct:
The u-lines passing through P and tangent
to c.
Constructions:
 Figure 20
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Let point O be Euclidean center
of c; and let A' be the reflection of A with respect to the
x-axis.
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Let S and S' be the intersections
of c and the circle defined by A, A' and O.
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Suppose that straight-lines
 and
 intersect
at P, construct the circle with diameter  which intersects
the given u-circle c at T and T'.
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Then T and T' are tangent
points; and u-lines AT and AT' are required tangents.
The target of this construction above is to
find out the circle(s) satisfying the following three conditions:
first, which passes through the point A, second, whose (Euclidean)
center lies on the x-axis, and third, which is tangent to the
given u-circle c.
According to the constructions above (see Figure 20), in the
circle c we have  ; and in the
circle c1 we have  ; so we have
 . Hence
 is tangent
to the circle defined by A, A' and T, which is just the required
circle.
We can see that: first, it passes through A;
second, it is tangent to the given u-circle c since they have
the common tangent  ; finally, its
(Euclidean) center lies on the x-axis since the x-axis is the
perpendicular bisector of its chord  .
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This Macro two_tangents_to_u_circle_from_point.mac
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