6 to 9 Point Circle: What Is This About?
A Mathematical Droodle

The existence of the 9-point circle for a given triangle, may be established in a variety of ways. In ΔABC, the 9-point circles passes through

M_a, M_b, M_c - the midpoints of sides BC, CA, and AB,
H_a, H_b, H_c - the feet of altitudes to sides BC, CA, and AB,
A_h, B_h, C_h - the midpoints of AH, BH, CH - the so called Euler points.

six point circle to nine point circle

Bui Quang Tuan came up with a novel proof based on the recent theorem of six concyclic points. Let, as usual, H denote the orthocenter of ΔABC. The proof begins with a construction of three circles (O_a), (O_b), (O_c), which are the circumcircles of triangles HH_aM_a, HH_bM_b, HH_cM_c. Of course the three centers O_a, O_b, O_c are the midpoints of HM_a, HM_b, HM_c. Two circles (O_b), (O_c) share one common point at H. Their other common point is the reflection of H in O_bO_c therefore it is on altitude AH_a. (This is because O_bO_c ⊥ HH_a.) Similarly we can show: three circles (O_a), (O_b), (O_c) share one common point at H and another common points are on the Cevians (altitudes) AH_a, BH_b, CH_c. By the six concyclic points theorem, H_a, H_b, H_c, M_a, M_b, M_c are concyclic on a circle, say, (N). In words,

In a given triangle, the feet of altitudes and the midpoints of the sides are concyclic.

Applying this statement to, say, ΔCAH we see that its circle (N) passes through the midpoints of its sides, viz., C_h, A_h, M_b and, of course, through the feet of the altitudes, which happen to be H_a, H_b, H_c. The circles (N) formed for triangles ABH also BCH, also pass through H_a, H_b, H_c so that all three coincide. The latter two pass through A_h, B_h, M_c and B_h, C_h, M_a, respectively. It follows that the common (N) circle passes through all the nine points.

Nine Point Circle

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6 to 9 Point Circle: What Is This About? A Mathematical Droodle

Nine Point Circle

6 to 9 Point Circle: What Is This About?
A Mathematical Droodle