32 III. PERPENDICULARS AND OBLIQUE LINE SEGMENTS
If we had started with an oblique segment OB less distant than OC, but on the
other side of H, it would suffice to construct a length HA = HB in the direction of
HC. The oblique segment OB then would be equal to OA
(2◦),
and hence smaller
than OC as we have seen above.
30. Conversely. If two oblique segments are equal, their feet are equally
distant from the foot of the perpendicular, otherwise they are unequal; if they are
unequal, the longer is the more distant from the perpendicular.
Corollary. There are no more than two oblique segments of the same length
from the same point O to a line xy.
This is true because the feet of these oblique segments are equally distant from
H, and there are only two points on xy at a given distance from H.
31. The length of the perpendicular dropped from a point to a line is called
the distance from the point to the line. The preceding theorem shows that this
perpendicular is in fact the shortest path from the point to the line.
32. Theorem.
1◦. Every point on the perpendicular bisector of a segment is equally distant
from the endpoints of the segment;
2◦. A point not on the perpendicular bisector is not equally distant from the
endpoints of the segment.
Figure 32
1◦.
If M is on the perpendicular bisector of AB (Fig. 32), the segments MA,
MB are equal, because they are oblique lines which are equally distant from the
foot of the perpendicular MO.
2◦.
Let M be a point not on the perpendicular bisector, and suppose it is
on the same side of the bisector as point B. The foot O of the perpendicular
dropped from M onto line AB will be on the same side of the bisector (otherwise
this perpendicular would meet the bisector and, from their intersection, there would
be two lines perpendicular to AB).
We will then have O A O B and therefore (29)
M A M B.
QED
Remark I. We could have proved the second part in a different way by estab-
lishing the following equivalent proposition: Any point which is equidistant from
A and B is on the perpendicular bisector of AB. This follows from the converse
of 30 (the feet of two equal oblique segments are equidistant from the foot of the
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