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Using Rolle's Theorem To Inscribe An Ellipse With Maximum Area In A Triangle Jeff Morgan Department of Mathematics University of Houston Definition: We say that a subset π of π 2 is an ellipse if and only if there are points π and π in π 2 , and a scalar πΏ > |π β π|, so that π is the set of points (π₯, π¦) in π 2 satisfying |(π₯, π¦) β π| + |(π₯, π¦) β π| = πΏ The points π and π in the definition are referred to as the foci of the ellipse, and the center of the ellipse is given by π· = 1 (π 2 + π) (the midpoint of the line segment ππ). Also, πΌ + π½ = πΏ. Theorem: The subset π in the π₯π¦ plane is an ellipse with center π· = (π₯0 , π¦0 ) if and only if there are scalars π, π and π so that π > 0, ππ β π 2 > 0, and π is the set of points (π₯, π¦) solving the equation π(π₯ β π₯0 )2 + π(π¦ β π¦0 )2 + 2π(π₯ β π₯0 )(π¦ β π¦0 ) = 1 The ElliptiGo 11R Bicycle β http://www.elliptigo.com/ Magnified view. Profile shot with graphic overlay. Geogebra Demo That was a big fail, but look at this demo? Does it produce an ellipse? Geogebra Demo β Perp Jig Here is another. Does it produce an ellipse? Geogebra Demo β NonPerp Jig OK. Enoughβ¦ We need some facts. Suppose π, π, π, π are real numbers and ππ β ππ β 0. If π is a π π subset of the π₯π¦ plane, then we define the action of ( ) on π π π as the set π’ {(π’, π£) | ( ) = (π π£ π π π₯ ) ( ) for some (π₯, π¦) β π} π π¦ 1 β2 Exercise: Give the action of ( ) on the line segment 1 1 connecting the points (β1,2) and (2, β3). Also, find the action 1 β2 of ( ) on the midpoint of this line segment, and comment 1 1 on any observations. π π ) on a line segment is a line π π segment, and the endpoints of the new line segment are the 1 β2 result of the action of ( ) on the endpoints of the original 1 1 line segment. In addition, the midpoint of the original line segment gets mapped to the midpoint of the new line segment. In General: The action of ( 1 β2 Exercise: Give the action of ( ) on the triangle with 1 1 vertices (β1,2), (2, β3) and (4,1). Also, compare the areas. In General: The action of ( π π π ) on a triangle π1 is a triangle π π2, and area(π2 ) = |ππ β ππ |area(π1 ) 1 β2 Exercise: Give the action of ( ) on the set consisting of 1 1 the parabola π¦ = π₯ 2 and its tangent line at π₯ = 1. Graph the results. In General: The action of a matrix on a set preserves tangent intersections of curves in the set. π More Fun Facts: The action of ( π ellipse πΈ2 , and π ) on an ellipse πΈ1 is an π area(πΈ2 ) = |ππ β ππ |area(πΈ1 ) A Simple Minimization Problem Find a triangle of minimum area for which a given circle of is an incircle. Our Featured Maximization Problem!! Find an ellipse of maximum area that is inscribed in a given triangle. Step 1: Note that maximizing the area of the ellipse is equivalent to maximizing area(ππππππ π) area(π‘πππππππ) Because the triangle is fixed. Step 2: Believe there is an ellipse that has maximum area. Now, take the given triangle and the ellipse (that you donβt have) and π π act a matrix ( ) on the combination so that the ellipse (that π π you donβt have) becomes a circle. What is area(ππππππ) ? area(πππ€ π‘πππππππ) Step 3: How does this help us solve our maximization problem? Hint: See the earlier minimization problem. Example: Find an ellipse of maximum area that is inscribed in the triangle with vertices (β1,2), (2, β3) and (4,1). A Much Simpler Solution and Comments on Rolleβs Theorem Example: Find an ellipse of maximum area that is inscribed in the triangle with vertices (β1,2), (2, β3) and (4,1). Step 1: Form the function π (π§) = (π§ β (β1 + 2π ))(π§ β (2 β 3π ))(π§ β (4 + π )) Step 2: Find the roots of πβ²(π§). Step 3: Find the roots of πβ²β²(π§). Step 4: How is this related to the solution? Terms: Steiner Inellipse, Mardenβs Theorem