In the above diagram we have the nine-point circle of triangle going through the midpoints
(of sides
respectively), and the feet of the altitudes
(from
in that order).
(1)
if (and only if) the side-lengths satisfy
(2)
It will also be shown, in our first example, that we have the relations
(3)
in any triangle (append absolute values if need be).
In the meantime, note that the right side of equation (1) evaluates to for a right triangle, where
is the circumradius, whereas both sides of equation (1) evaluate to
under (2). And so in the latter case, the segment
is not just a chord, but a diameter of the nine-point circle.
Let’s do this for an acute triangle. Slight modification for an obtuse triangle will be needed. By drawing an appropriate diagram we have:
We can re-write the expression for entirely in terms of the side-lengths using the cosine formula:
Similarly we obtain the expressions for and
.




Since , we have
by the usual notation. Using example 1 above we have
and so .

We first have . Moreover, if
is the circumradius, then
for a right triangle. Now:
Such a triangle is necessarily non-right. Let be its circumradius. By one of the equivalent statements here, we know that equation (2) then becomes equivalent to
. Re-arrange equation (2) in the form
and consider both sides of equation (1):

By example 4 above, we had . Since the nine-point circle goes through
and
and has radius equal to
, the fact that
means that the chord
is a diameter.
No other triangle has this property.
Takeaway
In triangle , let
denote the Euler point of vertex
,
the foot of the altitude from vertex
, and
the midpoint of side
. Then the following statements are equivalent:
- the chord
is a diameter of the nine-point circle
.
The equivalent conditions range from extremely simple to moderately involved.
Task
- (Late sixties) In a non-right triangle
, let
be the side-lengths,
the altitudes,
the feet of the altitudes from the respective vertices,
the midpoints of sides
,
the Euler points,
the circumradius,
the circumcenter,
the nine-point center,
the orthocenter,
the reflection of
over side
,
the reflection of
over side
, and
the reflection of
over side
. PROVE that the following sixty-eight statements are equivalent:
or
is congruent to
is congruent to
is isosceles with
is isosceles with
is right angled at
is the circumcenter of
is right-angled at
is right-angled at
- quadrilateral
is a rectangle
- the points
are concyclic with
as diameter
- the reflection
of
over
lies internally on
- the reflection
of
over
lies externally on
- radius
is parallel to side
is the reflection of
over side
- the nine-point center lies on
- the orthic triangle is isosceles with
- the geometric mean theorem holds
- the bisector of
has length
, where
- the orthocenter is a reflection of vertex
over side
- segment
is tangent to the circumcircle at point
- median
has the same length as the segment
- the bisector
of
is tangent to the nine-point circle at
is a convex kite with diagonals
and
- altitude
is tangent to the nine-point circle at
- chord
is a diameter of the nine-point circle
- segment
is tangent to the nine-point circle at
.
(short of the target. Next target now is to extend the initial target.)
- (Extra feature) If
satisfies equation (??), PROVE that its nine-point center
divides
in the ratio
.