Due to a little
-related issue, we chose to change the format of today’s post. Please click here for the pdf.

Takeaway
Let and
be the feet of the symmedian and the altitude (both) from vertex
in triangle
. Then the following statements are equivalent:
and
divide
harmonically
- the identity
holds.
Task
- (Growing membership) 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
in that order,
the Euler points,
the circumradius,
the circumcenter,
the nine-point center,
the orthocenter,
the reflection of
over side
,
the reflection of
over side
,
the reflection of
over side
,
the symmedian point,
the foot of the symmedian from vertex
, and
the radius of the polar circle. PROVE that the following eighty-four statements are equivalent:
or
and
divide
harmonically
is the reflection of
over side
is the reflection of
over side
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
- segment
is perpendicular to 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
.
(Once we hit, we’ll stop this particular exercise.)