Given three points A, B, C which are NOT collinear, a circle may be drawn passing through those points. The circle is centred at the point (O) where perpendicular bisectors of lines joining AB and BC intersect.
If we have more than three non-collinear points, they will not necessarily all lie on a circle but will sometimes do so. Thus, four points may form a CYCLIC QUADRILATERAL. For a quadrilateral to have all its vertices on a circle, the opposite angles must be supplementary.
A long-standing quest for mathematicians was to find more than four "triangle-related" points lying on the same circle. In 1765, Leonard Euler showed that six such points lie on a circle - i.e. the midpoints of the triangle's three sides and the feet of the three altitudes.
It was not until 1820 that an additional three points were added to Euler's six. The new points are the midpoints of segments from the orthocentre to the vertices. Hence, the NINE POINT CIRCLE.
In the next diagram, the blue points are the midpoints of the triangle sides. The red points are the feet of the altitudes. The green points are the midpoints of segments from the orthocentre to the vertices.
The nine-point circle centre, is the midpoint of the segment connecting the orthocentre and the circumcentre of a triangle. It lies on the Euler line, which also passes through the triangle's centroid and orthocentre.
The nine point circle is tangential to both the INCIRCLE and the EXCIRCLES of the triangle, This was discovered by Karl Wilhelm Feuerbach in 1822 and is known as Feuerbach's Theorem.
https://www.mayhematics.com/g/g3_ninepoints.htm
Nine Point Circle - Stonybrook Mathematics
Sam Hartburn - Nine Point Circle
Mayhematics.com - Nine Points Circle
Wikipedia - Nine Point Circle
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