The Green Zone Problem

I'd like to bring your attention to this easy geometry problem that I call "The Green Zone Problem". Let's suppose there is an imaginary city in the Middle-earth. I call this imaginary city as Indigo City (feel free to give any cool name) and its shape is a triangle like the picture below: The inscribed circle of the triangle is the incircle where the city council lives. The Green Zone is a city forest like Central Park in New York City. The interesting fact about the Green Zone is its boundary is always equidistant from the centre of circle and the closest triangle side, i.e. GO=DGGO=DG and JO=JK=JLJO=JK=JL but JOGOJO \neq GO. Using the conventional notations, we denote BC=aBC=a, AC=bAC=b, and AB=cAB=c. Just for the sake of fun, determine the area of the Green Zone in term of aa, bb, and cc.

Hint: If the shape of Indigo City is an equilateral triangle, unless I'm very much mistaken, then its area is 5a21083\dfrac{5a^2}{108}\sqrt{3}.

αρργρrσβlϵmsσlνιηg\color{blue}{\Huge-\hbar\alpha\rho\rho\gamma\,\,\,\rho r\sigma \beta l\epsilon m\,\,\,s\sigma l\nu \iota\eta g-}

Note by Tunk-Fey Ariawan
3 years, 1 month ago

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Ah, a nice way of introducing that wonderful property of X.

(Where X is to be determined by the reader)

Calvin Lin Staff - 3 years, 1 month ago

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For an equilateral triangle, I'm getting 2327a2\displaystyle \frac{2\sqrt{3}}{27}a^2, with aa being the side of the triangle. The boundaries of the green zone are made up of parabolas right? Unless I did something wrong here, then I have a method to generalise this.

This is the solution I got for an equilateral triangle:

Julian Poon - 2 years, 6 months ago

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