Notes
circles around a triangle solution

Circles Around a Triangle

Circles Around a Triangle

The diameters of these circles form a 3453-4-5 triangle. What’s the total shaded area?

Solution by Dissection and Area of a Triangle

Circles around a triangle labelled

With the regions labelled as above, they combine as follows:

Then the shaded region can be expressed as follows:

A+E+G+I+J =A+B+C =+E+F+G =+H+I =+D+J =BCFHD =L+T+S+M =BEF =CGF =DEFGH =+E+F+G+E+F+G =L+T+S+MMSL+T+T =3T \begin{aligned} A + E + G + I + J &= A + B + C \\ &\phantom{=} + E + F + G \\ &\phantom{=} + H + I \\ &\phantom{=} + D + J \\ &\phantom{=} - B - C - F - H - D \\ &= L + T + S + M \\ &\phantom{=} - B - E - F \\ &\phantom{=} - C - G - F \\ &\phantom{=} - D - E - F - G - H \\ &\phantom{=} + E + F + G + E + F + G \\ &= L + T + S + M - M - S - L + T + T \\ &= 3 T \end{aligned}

As the triangle has side lengths 33, 44, and 55 it is a right-angled triangle and has area 66. So the shaded regions have total area 1818.