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Prove that an isosceles triangle of maximum area that can be inscribed in an ellipse has its base parallel to one of the axes of the ellipse.

Note by Abhimanyu Swami
2 years, 6 months ago

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Here's a quickie intuitive explanation. Draw a circle with an inscribed equilateral triangle. This equilateral triangle has the maximum area any inscribed triangle can have. Draw the altitude which is perpendicular to the base. Now, if you stretch the circle and triangle together (you can scale the \(y\) component, for example). The altered triangle has the maximum area that can be inscribed in the altered circle which is now an ellipse. But note that the altitude and the base are no longer perpendicular, except in cases where one of the bases of the original equilateral triangle is either parallel or perpendicular to the stretch factor. Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin Owsome Explanation Mendrin sir Karan Shekhawat · 2 years, 5 months ago

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@Michael Mendrin Wow.. thanks. but can you also explain it using some coordinate geometry...? Abhimanyu Swami · 2 years, 6 months ago

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