**Intersecting Loci Principle In Geometric
Construction**

Najia Bao

I like to use the GeometerÕs Sketchpad, because it is a very helpful
tool. It allows us to investigate, demonstrate, and explore, specially to find
geometric loci.

As we know, the solution of a geometric
construction problem depends upon the finding of a point, which satisfies
certain conditions. In order to locate the point, we need two conditions, that
is to say, we can try to locate the point by finding the intersecting point of
two loci (lines or circles). We usually call the method Òintersecting lociÓ. If
we know lots of geometric loci, we can discover immediately where the required
point is to be located. But it is not easy to solve construction problem. Of course,
it is easy to find one locus, but what makes the problems difficult is that we
need to find another locus. GSP, as a means, helps us to visualize and
demonstrate.

Now we begin with the following problem.

*
*Problem 14*. Given a line and a circle
with center O. Take an arbitrary point P on the circle. Construct two circles
tangent to the given circle at P and tangent to the line. *

This is a difficult construction problem. Consider the given problem.
The center of the desired circle will lie along a line from the center *O *of the given circle with the specified point *P *(see fig. 1).

Figure 1.

But
this is just the first locus, which is easy to find. We have to find the second
locus to make *OÕP* = *OÕB*, and *OÕB*
be perpendicular to the line *l *(see
fig. 2).* *

Figure 2.

Now
we try to look for the other locus for the center *OÕ* of the tangent circle, which is difficult to find.
Suppose the problem had been solved, we would have the following figure (see
fig. 3):

Figure 3.

How
to find the second locus? As we know, the locus of a point equidistant from two
intersecting lines consists of the two bisectors of the angles formed by the
two given lines. We also know *OÕ *should
be the point of intersection of the two loci. Finally, a bisector of PAB formed by the two lines *l* and *l*_{2}.
*l*_{2 }is tangent to the
given circle at the point *P *(see
fig. 4).

Figure 4.

There are two solutions for this construction problem,
because not only PAB but also PABÕ are angles formed by the two lines *l* and *l*_{2.}
So we find the other circle whose center OÓ is the point of intersection of the
loci - line *l*_{1}nd *l*_{4}The following is the other solution (see
fig. 5):

Figure 5.

The above are my analysis and construction. Now I
want to give a proof and discussion on the construction problem.

*Proof*. Since OÕ lies on the bisector of *P**A**B*, *OÕP* ^ *l _{2}*, and

Similarly, we can prove that the circle with the center OÓ is also the
required circle.

*Discussion*. From the proof above, we conclude that
generally there are two solutions for this geometric construction problem. But there are two situations that there is only one
solution, when the given point *P *lies on a line which is perpendicular to the line *l
*and through the center *O* of the given circle. The following figures are the
two cases (see fig. 6 & fig. 7).

Figure 6.

Figure 7.

Whenever we do geometric
construction, we should keep in mind the most important and most frequently
useful geometric loci, such as the locus of a point equidistant from two given
points which is the perpendicular bisector of the given segment, and the locus
of a point equidistant from two intersecting lines indefinitely produced which
consists of the two bisectors of the angles formed by the two given lines. But,
for problems with difficult level, it is not enough to only depend on knowledge
of such geometric loci. We also need to apply a variety of problem solving
strategies and methods, such as dilation and tail construction. ApolloniusÕ
problems, PLL and PCC, the classical problems in geometric construction (see
Jim WilsonÕs Web site), can be solved by combining general geometric loci with
dilation (see fig. 8 & fig. 9).

PLL: Draw a circle passing through a given point
and tangent to two given lines.

Figure 8.

PCC: Draw a circle passing through a
given point and tangent to two given circles.

Figure 9