The Nine-Point Circle

Given a triangle ABC, construct the altitudes, which are concurrent at point H. Let the midpoints of the sides be A', B', C' as in the figure. Also, let A'', B'', C'' be the midpoints of HA, HB, HC.

Part 1. Shape of B'C'B''C''.

• Draw segments to form quadrilateral B'C'B''C''. What can you prove about this shape?

Suggestion: Make a drawing on the side only putting in the essential elements. What can you deduce.

Part 2. Consider the quadrilateral A'C'A''C'''.

What is the shape of A'C'A''C''? Does this require another proof?

Part 3. Adding a circle d

• In part 1, you found that B'C'B''C'' is a rectangle. This means that B'C'B''C'' can be inscribed in a circle d with B'B'' and C'C'' as diameters. Explain why this is true.

• In part 2, you found that A'C'A''C''is a rectangle. This means that A'C'A''C'' can be inscribed in a circle e with A'A'' and C'C'' as diameters. Explain why this is true.

• How are d and e related?

Part 4. First properties of circle d

• Circle d is the circumcircle of the midpoint triangle A'B'C'.

• Circle d is the circumcircle of the triangle A''B''C''.

• Recall that the orthocenter of triangle ABH is point C. Why?

• Conclude d is the circumcenter of the midpoint triangle of ABH (and also the circumcenter of the midpoint triangles of BCH and CAH).

Part 5. Adding the feet of the altitudes

Let DEF be the feet of the altitudes (as in the figure).

• D is on circle d. Why?

• E and F are also on circle d? Why?

Part 6. The Name of the circle d

Circle d is variously called the Nine-Point Circle of ABC or the Euler Circle of ABC.

What are the Nine Points?

Who was Euler?

Part 7. Dilations

Let c be the circumcircle of ABC. Then the circles c and d have two centers of dilation U and V that take circle c to circle d.

What is the center U for the dilation that takes c to d with positive ratio. What is the ratio? Explain.

What is the center V for the dilation that takes c to d with negative ratio. What is the ratio? Explain.

Part 8. Relations among the centers

Let O be the center of the circumcircle c and let N be the center of the nine-point circle d.

Putting these points together with "U" and "V" above, explain what these 4 points lie on a line and how they are situated relative to each other.