Medial Ellipse?

Geometry Level 5

Given a 3-4-5 triangle A B C ABC , find the area of the inscribed ellipse that is tangent to all three sides at their midpoints. If this area can be expressed as a π b c \frac{a\pi\sqrt{b}}{c} , where a a , b b and c c are positive, gcd ⁡ ( a , c ) = 1 \gcd(a,c)=1 and b b is square-free, find a + b + c a+b+c .


The answer is 8.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

link text
The ellipse we have is known as Steiner_inellipse. The above link gives its area as ,
A r e a o f S t e i n e r _ i n e l l i p s e = π 3 3 ∗ a r e a o f c i r c u m s c r i b i n g t r i a n g l e , = { π 3 3 } ∗ { 1 2 ∗ 3 ∗ 4 } = a π b c . ∴ a + b + c = 8 Area \ of\ Steiner\_inellipse\ =\dfrac \pi {3\sqrt3}*area\ of\ circumscribing\ triangle,\\ =\{\dfrac \pi {3\sqrt3}\}*\{\frac 1 2*3*4\}=\dfrac {a\pi \sqrt b} c.\\ \therefore\ \ a+b+c= \Large\ \ \color{#D61F06}{8}

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