

Next, a polar circle is a circle centred at the orthocenter of a triangle, with radius related to the circumradius
and the side-lengths
via
(1)
Now suppose that satisfy the identity:
(2)
Notice that any triangle in which equation (2) holds is necessarily obtuse, and so it’s polar circle is defined, with equation (1) simplifying to:
(3)
Not only this, it happens that precisely when
is the silver ratio.
Reduced attacks

The barrage was quite something prior to two weeks ago:

Grateful to find a barricade. It’s a different story now.
Refocused attention
Sorry we had to bother you with the above. Once in a while we’ll let you see what happens behind our scenes.


We’re to show that if , then the radius of the polar circle is
. Use the fact that in a non-right triangle, we have
.


We had from the preceding example. And also
. If we set
we get
after some simplification. In addition,
. Thus:
. Further simplification:
. Take the positive value:
, the silver ratio.


Similar to the preceding example.



We had . Since
, we then obtain
. Cosine formula:
.
Radical axis


Both the circumcircle and the polar circle have the same radius, under the given conditions. Thus the radical axis is the right bisector of the line joining their centers. But then the center of one is the circumcenter, while the center of the other is the orthocenter. The midpoint of these two centers is the nine-point center.
Takeaway
Suppose that triangle has side-lengths
, radius of polar circle
, and circumradius
. If
, then the following statements are equivalent:
is the silver ratio.
Tasks
- (Geometric mean) Suppose that the side-lengths of
satisfy
. PROVE that the following statements are equivalent:
is the silver ratio
- the nine-point center coincides with the foot of the symmedian from vertex
- the radius of the nine-point circle is the geometric mean of the two segments into which the nine-point center divides side
.
- (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-three statements are equivalent:
or
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
.
(Five more members were recently added.)