# THREE CONVEX HULL THEOREMS ON TRIANGLES AND CIRCLES

• Kalantari, Bahman (Department of Computer Science, Rutgers University) ;
• Park, Jong Youll (Department of Mathematics, Chonnam National University)
• Accepted : 2014.09.30
• Published : 2014.12.25
• 133 14

#### Abstract

We prove three convex hull theorems on triangles and circles. Given a triangle ${\triangle}$ and a point p, let ${\triangle}^{\prime}$ be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of ${\triangle}$. Let ${\triangle}^{{\prime}{\prime}}$ be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of ${\triangle}$. The first theorem characterizes when $p{\in}{\triangle}$ via a distance duality. The triangle algorithm in [1] utilizes a general version of this theorem to solve the convex hull membership problem in any dimension. The second theorem proves $p{\in}{\triangle}$ if and only if $p{\in}{\triangle}^{\prime}$. These are used to prove the third: Suppose p be does not lie on any extended edge of ${\triangle}$. Then $p{\in}{\triangle}$ if and only if $p{\in}{\triangle}^{{\prime{\prime}}$.

#### References

1. B. Kalantari, A characterization theorem and an algorithm for a convex hull problem, to appear in Annals of Operations Research, available online August, 2014. arxiv.org/pdf/1204.1873v2.pdf, and http://arxiv-web3.library.cornell.edu/pdf/1204.1873v4.pdf, 2012. To appear in Annals of Op erations Research, 2014.
2. B. Kalantari, Finding a lost treasure in convex hull of points from known distances, In the Proceedings of the 24th Canadian Conference on Computational Geometry (2012), 271-276.
3. B. Kalantari, Solving linear system of equations via a convex hull algorithm, arxiv.org/pdf/1210.7858v1.pdf, 2012.
4. B. Kalantari and M. Saks, On the Triangle Algorithm for the Convex Hull Membership, 2-page Extended Abstract, 23nd Annual Fall Workshop on Computational Geometry, City College of New York, 2013.
5. M. Li and B. Kalantari, Experimental Study of the Convex Hull Decision Problem via a New Geometric Algorithm, 2-page Extended Abstract, 23nd Annual Fall Workshop on Computational Geometry, City College of New York, 2013.
6. T. Gibson and B. Kalantari, Experiments with the Triangle Algorithm for Linear Systems, 2-page Extended Abstract, 23nd Annual Fall Workshop on Computational Geometry, City College of New York, 2013.
7. R. Johnson, A circle theorem, Amer. Math. Monthly 23 (1916), 161-162. https://doi.org/10.2307/2974356