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Re: to prove: rays bisecting 3 angles of a triangle meet at a single point
Posted:
Dec 28, 1997 5:34 AM
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Michael Keyton wrote:
> Do you know a property about how to tell if a point lies on the bisector
> of an angle, or not?
>
> If you know this theorem, apply it to two of the bisectors, in particular
> their intersection, and then argue that it is on the other bisector. And
> since an angle can only have one bisector, then the three meet at a point
> (the formal word for this phenomenon is "concurrent" and this point of
> concurrency is called the "incenter". It is one of over 100 centers of a
> triangle.
This incenter has this name because it is the center of the circle
inscribed in the triangle, i.e. touching the sides.
There are three very related points in a triangle, which can be found by
taking alternative bisectors for two of the angles. These alternative
bisectors can be found by lengthening the sides of the triangle,
resulting in a 'cross'. In this cross you see that there are in fact two
angles that can be bisected. The two alternative bisectors are
perpendicular.
new bisector
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\ | /
\ | /
\ | /
\ | /
\|/
-----X------ old bisector
/|\
/ | \
/ | \
/ | \
In this way you get three _excenters_, centers of three excircles. You
should try to find out what that means.
If you want to know more about the numerous triangle centers check out
the www-page of Clark Kimberling:
http://www.evansville.edu/~ck6/tcenters/index.html
Regards,
Floor van Lamoen
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