Then there’s this modification we want to make. Consider instead a circle having as diameter the line segment connecting the orthocenter with the circumcenter (the Euler line). This circle passes through vertex of a parent triangle
if, and only if, the side-lengths
satisfy
(1)
The reflection of the circumcenter over side also passes through this circle, under the above equivalence.















Let be the circumradius of the circle. In the diagram below
is the midpoint of
and
is the reflection of
over
, and so
. Since
and
, we have:







In the diagram below, is the midpoint of
, and so
is a median in triangle
.
So:
Similarly, is the midpoint of
, and again
is a median in triangle
:







This follows from the two preceding examples, since we now have and
.
(Main goal)









If equation (1) is satisfied, then from one of the equivalent statements here, the segment is perpendicuclar to the radius
.
Since the quadrilateral is a parallelogram, it follows that it is a rectangle. Consequently, the points
are concyclic. And the circle through these four points shares the same center with the nine-point center of the parent triangle.








Since is a diameter, the triangle
is right-angled at
. By one of the equivalent statements here, we conclude that equation (1) holds.
Takeaway
In a non-right , let
be the side-lengths,
the orthocenter,
the circumcenter, and
the reflection of the circumcenter over side
. Then the following statements are equivalent:
is right-angled at
is right-angled at
- quadrilateral
is a rectangle
- points
are concyclic with
as diameter.
Task
- (Half century) 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 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
- 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
.
(Quite plenty, but can we reach?)
- (Extra feature) If
satisfies equation (??), PROVE that its nine-point center
divides
in the ratio
.