The above property is exclusive to right triangles, and there are many other characterizations of right triangles, as seen in this list.
Today’s post will push at least two more entries into the list, in addition to that of July 14 .

Consider a right triangle in which
as shown below:
The length of the median to the hypotenuse is half the length of the hypotenuse: . Thus, the area is:

Suppose that the area of is equal to the product of the altitude
and median
from the same vertex
. Let
be the side-lengths and let
be the circumradius. Then
The area can be given as , and
itself can be written as
, by the extended law of sines. So:


Let be the circumradius of the parent triangle
. If the circle with diameter
coincides with the nine-point circle, then we must have
, since the radius of the nine-point circle is
.
the latter equation is one of the characterizations of right triangles given here.
In the next example we have a weaker requirement — we just ask that the nine-point circle pass through or
.
Suppose that the nine-point circle passes through . Let
be the nine-point center. Then
is a radius of the nine-point circle, namely
. But then
.
Similarly, if the nine-point circle of a triangle passes through the orthocenter of the triangle, then the parent triangle must be a right triangle.

of




The harmonic mean of and
is
. The square of the length of the bisector of
is given by
. We get
and then finally after simplifications.
Takeaway
In , let
be the side-lengths,
the median from vertex
,
the altitude from vertex
,
the orthocenter, and
the circumcenter. Then the following statements are equivalent:
- the area of
equals
is a right triangle with
- the nine-point circle of
passes through
- the nine-point circle of
passes through
- the circle with diameter
coincides with the nine-point circle of
is the geometric mean of the two equal segments it creates on the opposite side
- the square of the length of the bisector of angle
is half the square of the harmonic mean of
and
.
In the case of statement 6, one can replace the geometric mean with arithmetic mean or harmonic mean. Why?
Task
- (Early fifties) In a non-right triangle
, let
be the side-lengths,
the altitudes,
the feet of the altitudes from the respective vertices,
the circumradius,
the circumcenter,
the nine-point center,
the orthocenter,
the midpoint of side
, and
the reflection of
over side
. PROVE that the following fifty-four statements are equivalent:
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
- segment
is tangent to the nine-point circle at
.
(short of the target.)
- (Extra feature) If
satisfies equation (??), PROVE that its nine-point center
divides
in the ratio
.