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Obtuse Triangle

\begin{figure}\begin{center}\BoxedEPSF{ObtuseTriangle.epsf}\end{center}\end{figure}

An obtuse triangle is a Triangle in which one of the Angles is an Obtuse Angle. (Obviously, only a single Angle in a Triangle can be Obtuse or it wouldn't be a Triangle.) A triangle must be either obtuse, Acute, or Right.


A famous problem is to find the chance that three points picked randomly in a Plane are the Vertices of an obtuse triangle (Eisenberg and Sullivan 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the ``random'' points (Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the problem. Woolhouse (1886) solved the problem by picking uniformly distributed points in the unit Disk, and obtained

\begin{displaymath}
P_2=1-\left({{4\over\pi^2}-{1\over 8}}\right)= {9\over 8}-{4\over\pi^2}=0.719715\ldots.
\end{displaymath} (1)

The problem was generalized by Hall (1982) to $n$-D Ball Triangle Picking, and Buchta (1986) gave closed form evaluations for Hall's integrals.


\begin{figure}\begin{center}\BoxedEPSF{ObtuseTriangleArcs.epsf}\end{center}\end{figure}

Lewis Carroll (1893) posed and gave another solution to the problem as follows. Call the longest side of a Triangle $AB$, and call the Diameter $2r$. Draw arcs from $A$ and $B$ of Radius $2r$. Because the longest side of the Triangle is defined to be $AB$, the third Vertex of the Triangle must lie within the region $ABCA$. If the third Vertex lies within the Semicircle, the Triangle is an obtuse triangle. If the Vertex lies on the Semicircle (which will happen with probability 0), the Triangle is a Right Triangle. Otherwise, it is an Acute Triangle. The chance of obtaining an obtuse triangle is then the ratio of the Area of the Semicircle to that of $ABCA$. The Area of $ABCA$ is then twice the Area of a Sector minus the Area of the Triangle.

\begin{displaymath}
A_{\rm whole\ figure} = 2\left({4\pi r^2\over 6}\right)-\sqrt{3}\,r^2 = r^2({\textstyle{4\over 3}}\pi-\sqrt{3}\,).
\end{displaymath} (2)

Therefore,
\begin{displaymath}
P={{\textstyle{1\over 2}}\pi r^2\over r^2({\textstyle{4\over 3}}\pi-\sqrt{3}\,)} = {3\pi\over 8\pi-6\sqrt{3}}=0.63938\ldots.
\end{displaymath} (3)


Let the Vertices of a triangle in $n$-D be Normal (Gaussian) variates. The probability that a Gaussian triangle in $n$-D is obtuse is

$\displaystyle P_n$ $\textstyle =$ $\displaystyle {3\Gamma(n)\over\Gamma^2({\textstyle{1\over 2}}n)} \int_0^{1/3} {x^{(n-2)/2}\over(1+x)^n}\,dx$  
  $\textstyle =$ $\displaystyle {3\Gamma(n)\over\Gamma^2({\textstyle{1\over 2}}n)2^{n-1}}\int_0^{\pi/3} \sin^{n-1}\theta\,d\theta$  
  $\textstyle =$ $\displaystyle {6\Gamma(n)\,{}_2F_1({\textstyle{1\over 2}}n, n; 1+{\textstyle{1\...
...2}}n; -{\textstyle{1\over 3}})\over 3^{n/2}n\Gamma^2({\textstyle{1\over 2}}n)},$ (4)

where $\Gamma(n)$ is the Gamma Function and ${}_2F_1(a,b;c;x)$ is the Hypergeometric Function. For Even $n\equiv 2k$,
\begin{displaymath}
P_{2k}=3\sum_{j=k}^{2k-1}{2k-1\choose j}\left({1\over 4}\right)^j\left({3\over 4}\right)^{2k-1-j}
\end{displaymath} (5)

(Eisenberg and Sullivan 1996). The first few cases are explicitly
$\displaystyle P_2$ $\textstyle =$ $\displaystyle {\textstyle{3\over 4}}=0.75$ (6)
$\displaystyle P_3$ $\textstyle =$ $\displaystyle 1-{3\sqrt{3}\over 4\pi}=0.586503\ldots$ (7)
$\displaystyle P_4$ $\textstyle =$ $\displaystyle {\textstyle{15\over 32}}=0.46875$ (8)
$\displaystyle P_5$ $\textstyle =$ $\displaystyle 1-{9\sqrt{3}\over 8\pi}=0.37975499\ldots.$ (9)

See also Acute Angle, Acute Triangle, Ball Triangle Picking, Obtuse Angle, Right Triangle, Triangle


References

Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' Ill. J. Math. 30, 653-659, 1986.

Carroll, L. Pillow Problems & A Tangled Tale. New York: Dover, 1976.

Eisenberg, B. and Sullivan, R. ``Random Triangles $n$ Dimensions.'' Amer. Math. Monthly 103, 308-318, 1996.

Guy, R. K. ``There are Three Times as Many Obtuse-Angled Triangles as There are Acute-Angled Ones.'' Math. Mag. 66, 175-178, 1993.

Hall, G. R. ``Acute Triangles in the $n$-Ball.'' J. Appl. Prob. 19, 712-715, 1982.

Portnoy, S. ``A Lewis Carroll Pillow Problem: Probability on at Obtuse Triangle.'' Statist. Sci. 9, 279-284, 1994.

Wells, D. G. The Penguin Book of Interesting Puzzles. London: Penguin Books, pp. 67 and 248-249, 1992.

Woolhouse, W. S. B. Solution to Problem 1350. Mathematical Questions, with Their Solutions, from the Educational Times, 1. London: F. Hodgson and Son, 49-51, 1886.



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© 1996-9 Eric W. Weisstein
1999-05-26