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In the manipulative:

P1, P2, P3: foot of altitude

Q1, Q2, Q3: midpoint of each side

O1: orthocenter, i.e. the intersection point of three altitudes

O2: circumcenter, i.e. the intersection point of the bisector lines of three sides

R1, R2, R3: midpoint of the segments connecting each vertex and the orthocenter

Euler line: the segment orthocenter and the circumcenter

O: the midpoint of the the Euler line; also the center of the nine-point circle

The nine-point circle goes through P1, P2, P3, Q1, Q2, Q3, R1, R2, and R3.


Text below is from Wikipedia:

In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points, six lying on the triangle itself (unless the triangle is obtuse). They include:

  • The midpoint of each side of the triangle
  • The foot of each altitude
  • The midpoint of the segment of each altitude from its vertex to the orthocenter (where the three altitudes meet)

The nine-point circle is also known as Feuerbach's circle, Euler's circle, Terquem's circle, the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle.

The center of any nine-point circle (the nine-point center) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

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