The Nine-Point Circle
The Nine-Point Circle, also known as Euler's Circle or the Feuerbach Circle, is a figure that can be constructed using specific concyclic points defined by any given triangle.
The 9 points used to construct this circle are:
The midpoint of each side of the triangle
The foot of each altitude of the triangle (the point where each altitude intersects the opposite side at a right angle)
The midpoint of a line segment joining each vertex to the orthocenter of the triangle (where the orthocenter is the point of intersection of all three altitudes)
Acute vs. Obtuse Triangles
If the given triangle is acute, 6 of the 9 points (the 3 midpoints of the sides and the 3 altitude feet) will lie on the triangle itself.
If the given triangle is obtuse, only 4 of the 9 points will lie on the triangle itself. This happens because 2 of the altitude feet, the orthocenter, and all 3 midpoints between the orthocenter and the vertices will lie outside of the triangle. This leaves only 1 altitude foot and the 3 midpoints of the sides lying on the triangle itself.
Some interesting properties of the nine-point circle include:
Its radius is one-half the radius of the circumcircle of the given triangle.
Its center lies on the Euler line, at the midpoint between the triangle's orthocenter and circumcenter.
It is internally tangent to the triangle's incircle and externally tangent to the triangle's three excircles.
In the plot below, click to add 3 points defining a triangle. Then, click the button to follow through the steps of making the nine-point circle from your triangle.
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