
Assignment # 4 Centers of a Triangle by Victor
L. BrunaudVega 





The NinePoint Circle for any triangle
passes through: á The three midpoints of the sides; á The three feet of the altitudes; and á The three midpoints of the segments from the respective vertices to
the Orthocenter. 





Three points in the same plane but not
in the same line are common because that is the geometrical way to define a
circumference. Four points in
the same circumference are more rare.
But nine points! Come
on! Does it work for any
triangle? 


Well, this
animation says yes. If you
canŐt see it, maybe you should try here. 


What is the center of this circumference? 


To construct the middle points was easy, as well
finding the feet of the altitudes, but how can we find the center of the circumference?
We know the perpendicular bisectors of chords always go through the center of
the circle. So we can find the
center by connecting two pairs of points with line segments and intersecting
the perpendicular bisectors of the chords, as shown in the picture. 


Maybe is interesting to revise some
situations in which some points coincide collapsing and becoming not in nine
but in eight or less points. 



For example, if the triangle is
equilateral, the middle points of the sides coincide with the feet of the
altitudes of the triangle, so the circumference shares only three points with
the triangle. 
And if the triangle is isosceles,
the midpoint and the foot of the altitude of the base coincide, so the
circumference shares only eight points with the triangle as is shown in the
picture below. 






In a right triangle ABC, the feet
of the heights based on the legs coincide with the vertex where each leg
meets the hypotenuse. In this
case, we have only five common points between the triangle and the
circle. So, the center of the circle is in
the midpoint between the midpoint of the hypotenuse and the vertex of the
other two sides of the triangle, as you can see here. 




THE CENTER OF THE 9POINTS CIRCLE
IS THE MIDPOINT BETWEEN THE ORTHOCENTER AND THE CIRCUMCENTER. 


Another interesting characteristic
of this NinePoints Circle is that its center (N) is the middle point between
the Orthocenter and the circumcenter, as we can see in the picture and in this
animation. 



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