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Geometry · Circular sector · Isosceles right triangle · Tangent circles · Area subtraction

Problem 2, 2004

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In the isosceles right triangle \(ABC\) the right angle is at \(C\) and each leg has length \(2\). A circular arc \(\ell\) centred at \(A\) divides the triangle into two parts of equal area, and a circular arc \(m\) centred at \(B\) touches the arc \(\ell\) at a point of the hypotenuse \(AB\). What is the area of the shaded region?

A B C l m 45° 45° 2 2
The arcs \(\ell\) and \(m\) touch at a point of \(AB\); the shaded region is what is left of the triangle above them.

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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2004, 1. letnik, category A, problem 2. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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