





If we now reflect vertex over side
we obtain the circumcenter.
It follows, in the case of the -triangle, that the reflection triangle is degenerate; in addition, it degenerates to a special line connecting the centroid, circumcenter and orthocenter, known as the Euler line.
Is the -triangle the only triangle whose reflection triangle degenerates to the Euler line?





To see how this holds, we’ll show that the length of the altitude through
is half the distance from
to the orthocenter
. Indeed:
and
This shows that the reflection of vertex over side
is the orthocenter
.





Since , we have that
and
. Thus, the reflection of vertex
over side
is also the orthocenter.





To see this, note that since triangle is isosceles with
as the apex, the circumcenter lies on the axis of symmetry through
. We just have to show that the radius through
is twice the altitude through
. The altitude through
is:

We’ve already seen this by explicitly determining the reflection triangle from the preceding three examples. Alternatively, the necessary and sufficient condition for a degenerate reflection triangle is . This condition holds in the case of the
-triangle:

Let be the circumradius of the triangle. If the reflection of vertex
over side
is the orthocenter, then we must have
, as per the equivalent statements here. Similarly, if the reflection of vertex
over side
is the orthocenter, then we must have
. These two relations yield
. Since
is also equivalent to
, we obtain
. Thus, the triangle is the
-triangle.
Takeaway
In any triangle , let
be the side-lengths,
the orthocenter,
the circumcenter, and
the circumradius. Then the following statements are equivalent:
- the reflection of
over
is
- the reflection of
over
is
and the reflection of
over
is
Task
- (Aufbau) In triangle
, let
be the side-lengths,
the circumradius,
the circumcenter,
the nine-point center, and
the orthocenter. PROVE that the following statements are equivalent:
- the reflection of
over
is
- radius
is parallel to side
and radius
is parallel to side
- the reflection of
over
is
and the reflection of
over
is