Mathematics via Problems

Euler Line and Nine Point Circle

From Divine Proportions - Rational Trigonometry to Universal Geometry:

If $M_1,M_2$ and $M_3$ are the midpoints of the sides of the triangle $\overline{A_1{A}_2{A}_3}$, then $\overline{M_1{M}_2{M}_3}$ is also a triangle, and the circumcircle $n$ and circumcenter $N$ of $\overline{M_1{M}_2{M}_3}$ are respectively the nine point circle and the nine point center of the triangle $\overline{A_1{A}_2{A}_3}$.

Figure 18.1 shows the reason for the terminology, since in the decimal number field the nine point circle also passes through the feet $F_1,F_2$ and $F_3$ of the altitudes of $\overline{A_1{A}_2{A}_3}$, as well as the midpoints $P_1,P_2$ and $P_3$ of the respective sides $\overline{A_{1}O},\overline{A_{2}O}$ and $\overline{A_{3}O}$. Also shown in this Figure is the Euler line $e$.

How to draw the Nine-Point Circle of a given Triangle (9-Point Center)