Let E be the ellipse of maximum area that can be contained in a triangle of side lengths 1 , 3 , 2 .
Find the distance between the foci of the ellipse E .
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Sir, can you please post a solution for this: https://brilliant.org/problems/confusing-question-no-way-out/
A Steiner inellipse has the maximum area of any inellipse of a triangle, and for a triangle with sides a , b , and c it has semi-major and semi-minor axes 6 1 a 2 + b 2 + c 2 ± 2 Z where Z = a 4 + b 4 + c 4 − a 2 b 2 − b 2 c 2 − c 2 a 2 . Therefore, for a triangle with sides a = 1 , b = 3 , and c = 2 , the semi-major and semi-minor axes a e and b e are 6 1 8 ± 2 7 .
The distance d between the foci of an ellipse is twice its linear eccentricity c e , and c e = a e 2 − b e 2 . Therefore, d = 2 c e = 2 a e 2 − b e 2 = 2 ( 6 1 8 + 2 7 ) 2 − ( 6 1 8 − 2 7 ) 2 = 3 2 4 7 .
Sir, can you please post a solution for this: https://brilliant.org/problems/confusing-question-no-way-out/
Since solution does not compile in this space, I have to give it below.
The given formulas are from link below.
link text
T h i s i s a S t e i n e r E l l i p s e o f Δ w i t h s i d e s s a y a , b , c . T h e n E = 3 2 ∗ Z , w h e r e Z i s , Z 2 = a 4 + b 4 + c 4 − a 2 ∗ b 2 − b 2 ∗ c 2 − c 2 ∗ a 2 = 1 4 + ( 3 ) 4 + 2 4 − 1 2 ∗ 2 2 − ( 3 ) 2 ( 1 2 + 2 2 ) = 2 6 − 1 9 = 7 . ∴ E = 2 / 3 ∗ Z = 2 / 3 ∗ 4 7 .
What is the area of this Steiner inellipse ?
It is pi/6
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The Steiner inellipse has the maximum area of any inellipse of a triangle.
According to Marden's theorem , If the vertices of a triangle z 1 , z 2 , z 3 are the complex roots of a cubic polynomial f ( x ) , then the foci of the Steiner inellipse are the roots of the derivative f ′ ( x ) .
In this case letting z 1 = 0 , z 2 = 1 , z 3 = 3 i
We have f ( x ) = x ( x − 1 ) ( x − 3 i )
Letting f ′ ( x ) = 0 , we get 3 x 2 − 2 ( 1 + 3 i ) x + 3 i = 0
Now, the distance between the foci of the ellipse is equal to the absolute value of the difference between the roots of the above equation.
Which is ∣ ∣ ∣ ∣ 3 2 − 2 − 3 i ∣ ∣ ∣ ∣ = 3 2 4 7
Click here for the proof that the Steiner inellipse has the greatest area of all inellipses of a triangle.
Click here for the proof of Marden's theorem.