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7.Lemoine.
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- Nursing 220 Final Exam Review
Exam of 11 pages for the course Nursing 220 Final Exam Review at Nursing 220 Final Exam Review (7.Lemoine.)
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7. Lemoine Point.Two lines AS and AT through the vertex A of an an-
gle are said to be isogonal if they are equally inclined to
b or equivalently, to the bisector of A
the arms of A, b (Figure
1).
The isogonals of the medians of a triangle are called symmedians.
We will show in a little while that the symmedians are concurrent
and their point of concurrency is called the symmedian point. It
is also called the Lemoine point.
Figure 1:
Figure 2:
As before, in a triangle ABC, the midpoint BC is denoted by
b is A3 and then
A1 , the intersection of BC and the bisector of A
0
the symmedian of AA1 will be AA1 (Figure 2). Thus
AA01 = SymAA3 (AA1 ).
Recall Steiner’s theorem which states that in a tri-
angle ABC, if AA1 and AA2 are isogonal (Figure 3),
then
1
Figure 3:
, |AB|2 |BA1 ||BA2 |
= .
|AC|2 |CA1 ||CA2 |
We now apply this to get the following.
Theorem 1 A line AA01 in a triangle ABC
(Figure 4) is a symmedian if and only if
|BA01 | |AB|2 c2
= = .
|CA01 | |AC|2 b2
Figure 4:
Proof The line AA01 is a symmedian if AA1
is a median and
AA01 = SymAA3 (AA1 ).
Then |BA1 | = |CA1 |, so on applying Steiner’s theorem, we get that AA01 is
a symmedian if and only if
|AB|2 |BA01 ||BA1 | |BA01 |
= = .
|AC|2 |CA01 ||CA1 | |CA01 |
Remark It is well known that the bisector of an angle of a triangle
divides the opposite side into the ratio of the sides about the angle. Then,
be the above theorem, a symmedian does it in the ratio of the squares of the
sides.
2
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