October 24 – The Triangle Inequality

This Saturday we are very pleased to have as our returning speaker, Dr. Dorin Andrica.   He will share with us many applications of the Triangle Inequality which students will find useful on upcoming math contests.

Dorin Andrica received his Ph.D. in 1992 from “Babes-Bolyai” University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes-Bolyai” since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called “Andrica’s Conjecture.” He has been a regular faculty member at the Canada–USA Mathcamps between 2001–2005 and at the AwesomeMath Summer Program (AMSP) since 2006. Related articles on this site.

Please note that this meeting will be held in ECSS 2.311.

January 24, 2009: Geometry and Thebault’s Theorem

In anticipation for our speaker this Saturday I thought I would post what Wikipedia has to say about Thebault’s Theorem.   No preparation is required for each Math Circle lecture, but then again it can never hurt:

Thébault’s problem I

Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.

It is a special case of van Aubel’s theorem.

Thébault’s problem II

Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.

Thébault’s problem III

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.

Until 2003, acadamia thought this third problem of Thébault the most difficult to prove. It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that Y. Sawayama, an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.^[1]

Simion Filip – Geometry and Physics: from Euclid to Einstein

This Saturday, November 8th, Simion Filip will present the way our understanding of the physical world shaped the geometric problems that we considered throughout history.  Examples will be drawn mostly from elementary Euclidian geometry and the talk should be accessible to anyone who is familiar with angles and triangles.

Mr. Filip is a senior at Princeton University, studying mathematics with an interest in mathematical physics. After graduation, he plans to pursue a Ph. D. in mathematics.  While in high-school, Simion Filip took part in both the International Mathematics and Informatics Olympiads, where he received silver and bronze medals respectively.

Chengde Feng – Angles and Areas – Recap

Chengde Feng’s lecture was greeted with very high attendance and very active participation.  Throughout his lecture, Mr. Feng showed the students how to solve a wide variety of geometry problems involving area.  These are problems that frequently occur in contests like MATHCOUNTS or AMC.  Many shortcuts and techniques were shared to help shave off precious sections during these problem solving tests.

You knew you were at a Math Circle when in the second hour, Mr. Feng announced a “pop quiz” and instead of groans there was honest enthusiasm from the students eager to apply their new knowledge.

Mr. Feng has provided the problems from the lecture. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below.

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Chengde Feng – Angles and Areas – November 1

Chengde Feng, an instructor at AwesomeMath, father of Zuming Feng and a good friend of the Metroplex Math Circle will come to Dallas this Saturday to lecture on “Angles and Areas.”  To help us prepare, Mr. Feng has sent along this list of facts that every Middle School Student should know.

Concepts You Need to Know

1.  Angles of a Triangle

• Vertical angles are congruent
• The sum of the measures of the angles of a triangle is 180.
• The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

2. Angles of a Polygon

• The sum of the measures of the interior angles of a n-gon is
• The measure of each interior angle of a regular n-gon is
• The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

3. Triangles

• The length of each side of a triangle must be less than the sum of the lengths of the other two sides.
• If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
• An equilateral triangle has three angles.
• Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

4.  Areas The area of:

• a rectangle equals the product of its base and height.
• a parallelogram equals the product of its base and height.
• a triangle equals half the product of its base and height.
• an equilateral triangle with side length l is
• a rhombus equals half of the product of its diagonals.
• a trapezoid equals half the product of the height and the sum of the bases.
• a circle with radius r is and its circumference is .
• a sector AOB of circle O with radius r is

Minimal Surfaces and Regular Polyhedron – Recap

Alicia Prieto Langarica continued her tradition of actively engaging the students as they explored deep mathematical concepts. Ms. Langarica began with a discussion of the regular polyhedron and allowed the students to prove for themselves why there can be no more than 5. She then talked about these 5 Platonic Solids and gave some of the cultural context of these important objects.

From this basis, Ms. Langarica was able to describe the wide variety of non-regular polyhedron. As diverse as these objects are, they all have the common properties described by Euler. Ms. Langarica showed how the relationship between the number of vertices, faces and sides would be constant for all of these figures.

To involve the students more directly, the students built their own polyhedron and demonstrated their own diversity and talent. This break activity prepared them for listening to the more challenging portion of the lecture on Minimal Surfaces. Ms. Langarica described the very practical value of finding minimal surfaces to conserve cost or weight in construction projects. She then showed several beautiful examples of minimal surfaces.

By finding the surface normal of any point on a curved shape, Ms. Langarica showed how a minimal surface could be tested or created. To drive this point home, Ms. Langarica took the math circle outside with their polyhedron creations. By dipping these objects into soap bubbles she was able to beautifully demonstrate how minimal surfaces would form spontaneously as a result of the physical properties of the air and soap film.

Ms. Lanagrica has provided her slides from the lecture and answers to the problems. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below.

Click to join MetroplexMathCircle

Minimal Surfaces and Regular Polyhedra

October 11, we will begin with an introduction to polyhedrons and their properties. We will see why there is only a certain number of regular polyhedrons and we will talk about non regular polyherons as well. We will build our own polyhedrons, regular and non regular. Then we will introduce the concept of minimal surfaces and discuss its importance in various fields. The last activity is going to be a fun surprise that will illustrate minimal surface on different shapes.

Ms. Langarica is a mathematics Phd student at The University of Texas at Arlington. She has a BS in Applied Mathematics from the Univestity of Texas at Dallas and was a contestant in the Mexican Mathematics Olympiads for 5 years where she received one national silver medal and two gold medals. Since then, she has been involved continuously in mathematics Olympiads as a trainer and problem writer.

Introduction to Geometry – Richard Rusczyk

Our September 20th, 2008 speaker, Richard Rusczyk, is the author of several books designed to help teenage problem solvers. Introduction to Geometry is an excellent text that many of our Math Circle participants have used, often along with the Introduction to Geometry course at Art of Problem Solving.

Learn the fundamentals of geometry from former USA Mathematical Olympiad winner Richard Rusczyk. Topics covered in the book include similar triangles, congruent triangles, quadrilaterals, polygons, circles, funky areas, power of a point, three-dimensional geometry, transformations, and much more.

…the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which geometric techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains over 900 problems. The solutions manual contains full solutions to all of the problems, not just answers.