BrainDen.com - Brain Teasers

## Question

What is the median area found between three tangential circles when the radius of the three circles are between 1inch and 5 inch inclusive?

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The three radii added pairwise are the sides of a triangle defined by the centers of the circles.

The angles are given by law of cosines.

The chords are given by the radii and angles, as are the segment areas.

The area of the triangle formed by the three points of tangency is given by the chord lengths and Heron's formula.

The area bounded by the circles is that area minus the three segment areas.

I let each radius take the values 1,2,3,4,5 to obtain 125 bounded areas.

These ranged froim 0.161254 for (1,1,1) to 4.03162 for (5,5,5).

The median value (number 63 in the ordered list) was found to be 0.95248.

Was the puzzle to find how to calculate the area?
Is the result particularly interesting?
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Assuming yes to both questions:

Assign a range of values to the radii.

Let each have discrete values equally spaced over the interval [1, 5].

For example 1.5, 2.5, 3.5, 4.5, which would give 64 cases.

Compute the areas and find the median value.

The circles' centers define a triangle whose sides are pairwise sums of the radii.

From the sides calculate the triangle's area.

Get the three angles from the law of cosines.

Each circle has a sector of area  A r2/2 where A is the angle for that circle.

Subtract the area of the three sectors from that of the triangle.

It's NOT the area for three circles having median radius of 3.
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sometimes... learning new techniques to solving the familiar is the only award one gets.  The result means little to me.  I was just interested in seeing how others attacked this problem.

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FWIW:

The median area of 0.95248  was obtained for radii of (2 2 4) inches.

The next larger area 0.955875 was obtained for radii of (1 1 5) inches.

The next smaller area 0.858358 was obtained for radii of (1 4 5) inches.

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If we use non-integer radii does the inner area approach an irrational number?

With probability 1 it is irrational for circles of any radii -- integer or otherwise.

The area of a unit circle is pi.

The included area of three unit circles involves sin(60o).

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