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On an Elementary Proof of The Pancake Theorem a)Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia Abstract The Pancake Theorem states that for any two bounded region on the plane, there is a line that divides these two region of equal area simultaneously. This theorem is a special case of the Ham-Sandwich Theorem for two dimensional case. The Ham-Sandwich Theorem is usually proof by the uses of the Borsuk-Ulam Theorem and it is not accessible for high school and the first-year university student. Fortunately, there is an elementary proof of the Pancake Theorem as an application of the Intermediate Value Theorem for continuous function on the closed and bounded interval. This proof can be found as an exercise in several Calculus textbooks. However, the proof of the continuity of the function in this proof is usually omitted. In this paper, we investigate several cases of the Pancake Theorem when the continuity argument in its elementary proof can be verified. We also examine an elementary proof of Pancake Theorem for special regions such as rectangle, circle, quadrilateral, regular pentagon, regular hexagon and generalization for irregular plane figure. Our result can be viewed as an example of interesting application of continuous function and high school geometry. Keywords: Application of Intermediate Value Theorem, The Pancake Theorem, Application of Continous Function, The Ham-Sandwich Theorem, The Borsuk-Ulam Theorem Proof Topic: MATHEMATICS AND SCIENCE EDUCATIONS |
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