Writing for the third time on the geometric mean theorem, after an external eye scrutinized and stamped the previous two writings on the same theme, seems a joy thing.
With that said, today’s post focuses on numerical problems that concretize the geometric mean theorem in obtuse triangles.
Wow! Thanks for clicking.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com A(-4,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-bc3f15725e0a4923e8ba6e2df2ad7df3_l3.png)
![Rendered by QuickLaTeX.com B(0,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-7045877de833cb7e53ba1ef4ddfaeb24_l3.png)
![Rendered by QuickLaTeX.com C(2,2\sqrt{3})](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-51f4dd5600aa397f923da5141eeb676f_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com CF_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-40738933ec670187c6d5090a43e41261_l3.png)
![Rendered by QuickLaTeX.com AF_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-e20ed9e709286fbf98a34633245b83e0_l3.png)
![Rendered by QuickLaTeX.com BF_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4936771b3e1d7955c46cb2ad88b9cce7_l3.png)
Note that the triangle in this first example is a 30-30-120 triangle; it satisfies the geometric mean theorem twice.
From the above diagram, . Also
and
. Thus:
Similarly, if denotes the foot of the altitude from vertex
, then:
Another unique property of the triangle.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com A(0,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-a4c057a4f58741a4e3fc4d5121ae3540_l3.png)
![Rendered by QuickLaTeX.com B(3,6)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-0b67631b2dbdedc7cc3d349be4e9a2d8_l3.png)
![Rendered by QuickLaTeX.com C(0,10)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f5c0cc9017e2a098d03651147c899333_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com CF_c=\sqrt{AF_c\times BF_c}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-1eac26a6af99c13662c402b46004ae73_l3.png)
From the above diagram, . And so:
In words: the altitude is the geometric mean of the segments
and
.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com A(0,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-a4c057a4f58741a4e3fc4d5121ae3540_l3.png)
![Rendered by QuickLaTeX.com B(3,-9)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-43212633f3c663edcc119dfc80752010_l3.png)
![Rendered by QuickLaTeX.com C(5,5)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-5bae280ade7d95e5b5374fdea52bf0a7_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com CF_c=\sqrt{AF_c\times BF_c}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-1eac26a6af99c13662c402b46004ae73_l3.png)
From the above diagram, . And so:
In words: the altitude is the geometric mean of the segments
and
.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com A(-6,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-79c16eb208dc820695dfd9a30fa1c847_l3.png)
![Rendered by QuickLaTeX.com B(0,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-7045877de833cb7e53ba1ef4ddfaeb24_l3.png)
![Rendered by QuickLaTeX.com C(2,4)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-e2fb9eee47c72cdf6ab23e648c29df8c_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com CF_c=\sqrt{AF_c\times BF_c}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-1eac26a6af99c13662c402b46004ae73_l3.png)
From the above diagram, . And so:
In words: the altitude is the geometric mean of the segments
and
.
![Rendered by QuickLaTeX.com \triangle ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b8aadc08f11d62c883914b8be4436356_l3.png)
![Rendered by QuickLaTeX.com A(4,4)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4acc2a75b5891b8bc98456dcbc05edc4_l3.png)
![Rendered by QuickLaTeX.com B(0,0)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-7045877de833cb7e53ba1ef4ddfaeb24_l3.png)
![Rendered by QuickLaTeX.com C(1,-2)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c0508a321bf286d7f4b8683a94a1694c_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com CF_c=\sqrt{AF_c\times BF_c}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-1eac26a6af99c13662c402b46004ae73_l3.png)
From the above diagram, . And so:
In words: the altitude is the geometric mean of the segments
and
.
Takeaway
The two statements below, the geometric mean theorem, and (more than) 75 other statements, are equivalent in any non-right triangle with circumradius and side-lenths
:
.
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
- the circle with diameter
passes through vertices
and
- 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.)