Nine Point Circle: What Is This About?
A Mathematical Droodle

Explanation

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First of all, M_BM_C is the midline in ΔABC so that it's parallel to BC and equals half of the latter. The same holds of B_HC_H which is a midline in ΔHBC. It follows that M_BM_CB_HC_H is a parallelogram. But more is true. M_BC_H is a midline in ΔAHC. In particular, this implies that M_BC_H is orthogonal to BC (as it is parallel to AH.) Hence the quadrilateral M_BM_CB_HC_H is a rectangle.

Observe that the diagonals of a rectangle (of a parallelogram in fact) cross at their midpoints. Let N be the center of the rectangle M_BM_CB_HC_H.

Rectangles M_CM_AC_HA_H and M_AM_BA_HB_H are obtained in a similar way. Between them, the three rectangles share three diagonals: M_AA_H, M_BB_H, M_CC_H and therefore have a common center. This shows that 6 points - M_A, M_B, M_C, A_H, B_H, C_H lie on a circle with center N.

Furthermore, in ΔA_HH_AM_A the angle at H_A is right whereas the hypotenuse A_HM_A serves as a diameter of the just found circle. It follows that H_A also lies on that circle, and similar argument applies to the feet of the remaining altitudes, H_B and H_C.

(Note: Another simple argument can be found elsewhere at the site.)

Nine Point Circle

• Nine Point Circle: an Elementary Proof
• Feuerbach's Theorem
• Feuerbach's Theorem: a Proof
• Four 9-Point Circles in a Quadrilateral
• Four Triangles, One Circle
• Hart Circle
• Incidence in Feuerbach's Theorem
• Six Point Circle
• Nine Point Circle
• 6 to 9 Point Circle
  • Six Concyclic Points
• Six Concyclic Points II
• Bevan's Point and Theorem

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