A follow-up/corollary to our previous post: yet another characterization of the triangle whose nine-point center coincides with one of its vertices, namely the
-triangle. pdf here.
![Rendered by QuickLaTeX.com 30^{\circ}-30^{\circ}-120^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-0a10497c2335a5cd89bea71b5766548a_l3.png)
In
, let
and
. PROVE that the nine-point circle of
coincides with the incircle of
, where
is the orthocenter.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com \angle A=\angle C=30^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-408e55b18738970715ba119fa33d44b9_l3.png)
![Rendered by QuickLaTeX.com \angle B=120^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-09aa04cb2daf52ea1c2a9cb2b19f19f8_l3.png)
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com \triangle ACH](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c09051ea2fb08a3995cf7e62a859af3e_l3.png)
![Rendered by QuickLaTeX.com H](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-d2121613062e686e294d3867f5da2955_l3.png)
See the pdf here.
Suppose that the nine-point circle of
coincides with the incircle of
, where
is the orthocenter. PROVE that
and
.
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com \triangle ACH](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c09051ea2fb08a3995cf7e62a859af3e_l3.png)
![Rendered by QuickLaTeX.com H](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-d2121613062e686e294d3867f5da2955_l3.png)
![Rendered by QuickLaTeX.com \angle A=\angle C=30^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-408e55b18738970715ba119fa33d44b9_l3.png)
![Rendered by QuickLaTeX.com \angle B=120^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-09aa04cb2daf52ea1c2a9cb2b19f19f8_l3.png)
Same link as above: pdf here.
Takeaway
In any triangle , let
be the side-lengths,
the orthocenter,
the circumcenter, and
the circumradius. Then the following statements are equivalent:
is equilateral
- the reflection of
over
is
- the nine-point circle of
is the same as the incircle of
- 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:
is equilateral
- the reflection of
over
is
- the area
satisfies
- the circle with diameter
passes through vertices
and
- the nine-point circle of
is the same as the incircle of
- radius
is parallel to side
and radius
is parallel to side
- the reflection of
over
is
and the reflection of
over
is
is the geometric mean of
and
, and
is the geometric mean of
and
. (
and
are the feet of the altitudes from
and
, respectively.)