Exam (elaborations)
7.Lemoine.
- Institution
- 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.)
[Show more]
Seller
Follow
Content preview
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller modockochieng06. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £6.48. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
62890 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now