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Question Number 66211 by mr W last updated on 10/Aug/19

Commented by mr W last updated on 10/Aug/19

Find the maximum area of a right  triangle inscribed in an ellipse.

$${Find}\:{the}\:{maximum}\:{area}\:{of}\:{a}\:{right} \\ $$$${triangle}\:{inscribed}\:{in}\:{an}\:{ellipse}. \\ $$

Commented by MJS last updated on 10/Aug/19

I have got no time right now, but the centroid  of the triangle must be the center of the  ellipse.  I once posted the equations of the ellipse  with minimal area surrounding any triangle  it should be possible to reverse this system  to find the family of triangles with maximal  area inscribed in any ellipse (=all triangles  with vertices on the ellipse and centroid in  the center of the ellipse) and to find the  right angled one(s) out of this family

$$\mathrm{I}\:\mathrm{have}\:\mathrm{got}\:\mathrm{no}\:\mathrm{time}\:\mathrm{right}\:\mathrm{now},\:\mathrm{but}\:\mathrm{the}\:\mathrm{centroid} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{must}\:\mathrm{be}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{ellipse}. \\ $$$$\mathrm{I}\:\mathrm{once}\:\mathrm{posted}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse} \\ $$$$\mathrm{with}\:\mathrm{minimal}\:\mathrm{area}\:\mathrm{surrounding}\:\mathrm{any}\:\mathrm{triangle} \\ $$$$\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{reverse}\:\mathrm{this}\:\mathrm{system} \\ $$$$\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{family}\:\mathrm{of}\:\mathrm{triangles}\:\mathrm{with}\:\mathrm{maximal} \\ $$$$\mathrm{area}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{any}\:\mathrm{ellipse}\:\left(=\mathrm{all}\:\mathrm{triangles}\right. \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ellipse}\:\mathrm{and}\:\mathrm{centroid}\:\mathrm{in} \\ $$$$\left.\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ellipse}\right)\:\mathrm{and}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{right}\:\mathrm{angled}\:\mathrm{one}\left(\mathrm{s}\right)\:\mathrm{out}\:\mathrm{of}\:\mathrm{this}\:\mathrm{family} \\ $$

Commented by mr W last updated on 11/Aug/19

thank you sir!  can we say that the maximum triangle  is always isosceles with C on the  major or minor axis?

$${thank}\:{you}\:{sir}! \\ $$$${can}\:{we}\:{say}\:{that}\:{the}\:{maximum}\:{triangle} \\ $$$${is}\:{always}\:{isosceles}\:{with}\:{C}\:{on}\:{the} \\ $$$${major}\:{or}\:{minor}\:{axis}? \\ $$

Commented by MJS last updated on 11/Aug/19

draw a circle with radius a  inscribe an equilateral triangle with C= ((0),(a) )  rotate it by any angle θ  transform the whole thing: P= ((x),(y) ) → P′= ((x),(((by)/a)) )  all possible triangles have the same area  which is the max. area of triangles inscribed  in the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1

$$\mathrm{draw}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{radius}\:{a} \\ $$$$\mathrm{inscribe}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with}\:{C}=\begin{pmatrix}{\mathrm{0}}\\{{a}}\end{pmatrix} \\ $$$$\mathrm{rotate}\:\mathrm{it}\:\mathrm{by}\:\mathrm{any}\:\mathrm{angle}\:\theta \\ $$$$\mathrm{transform}\:\mathrm{the}\:\mathrm{whole}\:\mathrm{thing}:\:{P}=\begin{pmatrix}{{x}}\\{{y}}\end{pmatrix}\:\rightarrow\:{P}'=\begin{pmatrix}{{x}}\\{\frac{{by}}{{a}}}\end{pmatrix} \\ $$$$\mathrm{all}\:\mathrm{possible}\:\mathrm{triangles}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{area} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{the}\:\mathrm{max}.\:\mathrm{area}\:\mathrm{of}\:\mathrm{triangles}\:\mathrm{inscribed} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$

Answered by MJS last updated on 11/Aug/19

Commented by mr W last updated on 12/Aug/19

thanks again sir!

$${thanks}\:{again}\:{sir}! \\ $$

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