| Some related info:
Because of the three-way symmetry of triangles, the idea that
a three sided figure determines two points, seems at first sight
as puzzling as that lines which cut off thirds from the sides should
cut out one-seventh of the area.
The points were first brought to the attention of mathematicians
by H. Brocard (1845-1922), and they have received a great deal of
study and found to possess many interesting properties.
The "2-ness" comes from the fact that a line to a vertex makes an
angle clockwise to one side and counter-clockwise to the other.
A Brocard point connected with the three vertices makes the alternate
set of angles, all clockwise or all counter-clockwise, equal.
Some questions:
1. For a 3-4-5 triangle find the Brocard points.
2. For a triangle whose sides are 13, 14, 15 and one altitute
is 12, the area is 84, the sum of the squares of the sides
is 590, find the Brocard points.
/Enjoy,
Kostas
|
| Three (non-colinear) points determine a circle, and this determines
two points: the centers of the inscribed and circumscribed circles.
In fact, you can choose any two singular points determined by the
three given points:
o Center of the inscribed circle
o Center of the circumscribed circle
o Mean of the three points (average the x-y coordinates).
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