Circle geometry

(FB/Math/BM – Proposed by Souradeep Purkayastha)

Three circles touch each other, and two lines are direct common tangents to the three circles. Apart from the three, there are of course many more tangent circles that can be drawn to continue the series. Now, show that the radii of the circles forms a mathematical progression.

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Let the distance from the intersection of the lines to the centre of a circle be d_n, and the radius r_n. Then from similar triangles and
dn+1 = dn + rn+1 + rn, we have

sin v = rn+1/dn+1 = rn/dn = (rn+1 – rn)/(dn+1 – dn) = = (rn+1 – rn)/(rn+1 + rn),

which gives

rn+1/rn – 1 = (rn+1/rn + 1)·sin v

rn+1/rn = (1 + sin v)/(1 – sin v)

so the ratio rn+1/rn is constant. Thus the radii form a geometric progression. Note that v is half angle between the lines.

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