A triangle in the middle

Geometry Level 2

3 3 circles of radius 1 , 2 , 1, 2, and 3 3 abut up against each other as shown:

What is the area of the triangle formed by joining the centers of the circles?


The answer is 6.

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Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Apr 4, 2017

Since the radii are 1,2, and 3, the triangle defined by their centers is a 3-4-5 triangle (a right triangle) so its area is given by:

Area = 3 ⋅ 4 2 = \frac{3\cdot 4}{2}

Area = 6 \boxed6

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Curiously, if the triangle formed is a right triangle, then its inradius is equal to the radius of the smallest of the circles.

Suppose the three radii a < b < c a \lt b \lt c are such that

( a + b ) 2 + ( a + c ) 2 = ( b + c ) 2 ⟹ (a + b)^{2} + (a + c)^{2} = (b + c)^{2} \Longrightarrow

2 a 2 + 2 a b + 2 a c + b 2 + c 2 = b 2 + 2 b c + c 2 ⟹ a ( a + b + c ) = b c 2a^{2} + 2ab + 2ac + b^{2} + c^{2} = b^{2} + 2bc + c^{2} \Longrightarrow a(a + b + c) = bc .

Now the inradius r = A s = ( a + b ) ( a + c ) 2 a + b + c = a ( a + b + c ) + b c 2 ( a + b + c ) = a r = \dfrac{A}{s} = \dfrac{\dfrac{(a + b)(a + c)}{2}}{a + b + c} = \dfrac{a(a + b + c) + bc}{2(a + b + c)} = a .

Brian Charlesworth - 4 years, 2 months ago

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