If we now reflect the circumcenter over the sides of the triangle, the reflections do not lie on the circumcircle, unless a certain condition is met.
Sixty degrees
In the presence of an interior angle of or
, the reflection of the circumcenter lies on the circumcircle.







Let be the circumradius of the triangle, and let
be its orthocenter. In the diagram below
suppose that actually lies on the circumcircle. Then
must equal
, right? We showed previously that
in any triangle with orthocenter
. But then
always equals
. And so:




As before, let’s denote by the reflection of the circumcenter
over side
. We’ll establish the result for the
case.
The angle which the minor arc through and
subtends at the center is equal to
; that is, obtuse
, and so reflex
. Note that
is a rhombus, and so obtuse
.
Since reflex , we conclude that
lies on the circumcircle.
Simple deduction
As a consequence of the preceding examples, we can characterize equilateral triangles via reflections of the circumcenter.
Easy.
Sample diagrams




Note that this is a right triangle in which ,
, and
. The circumcenter is the point
, and its reflection over side
is the point
.




Here, , and
. An extremely pleasant case.
Takeaway
In , let
be the circumcenter, and let
be the reflection of
over side
. Then the following statements are equivalent:
.
And a few more, if you view this post.
Task
- (Late 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
,
the reflection of
over side
,
the reflection of
over side
, and
the reflection of
over side
. PROVE that the following fifty-seven statements are equivalent:
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
- 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
.