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To construct the unique u-line that is perpendicular
to two given (ultra parallel) u-lines is equivalent to construct
the circle that is orthogonal to two given circles and with its
center on the line of centers (see Figure 13).
 Figure 13
We can observe that the center of the circle
p is equal power with respect to the two given circles r1 and
r2. By the Chordal Theorem, the locus of points of equal
power with respect to two non-concentric circles is a straight-line
that is perpendicular to the line of centers, and is called the
‘radical line’ of the two given circles.
 Figure 14
Let q be an arbitrary circle which intersects r1 and r2 at A,
B, C, D (see Figure 14). Suppose lines  and
 intersect
at Q, then Q is equal power with respect to r1 and r2. Since the
power of Q with respect to r1 is equal to  , which is also
the power of Q with respect to q, and so is  , the power of
Q with respect of r2.
The line, which is perpendicular to the line
of centers and passes through Q, is hence the ‘radical line’.
The projection of Q on the line of centers (labeled as P) is hence
the center of the common orthogonal circle p in Figure 13.
Since an arbitrary circle does not necessarily
intersect the two given circles, in the following construction
we will use a fixed circle instead.
Given:
Two (ultra parallel) u-lines m and n in the
upper half-plane.
To Construct:
The unique u-line that is perpendicular to
the two given u-lines.
Constructions:
 Figure 15
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Let C1 and C2 be centers of circles consisting
of u-lines m and n respectively. Let
 and
 be perpendicular
lines to the x-axis and intersect m and n at B1 and B2 (see
Figure 15).
-
Let M be the midpoint of B1 and B2. Let lines
 and
 be perpendiculars
to  and
 respectively
and intersect at N. (Note that the circle with diameter
 is the
fixed circle mentioned above.)
-
Let P be foot of perpendicular of N to the
x-axis. Draw the circle with diameter
 and intersect
m at Q. Then the u-line through Q with Euclidean center P
cuts u-lines m and n orthogonally.
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This Macro perpendicular_to_two_u_lines.mac
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