Three

**circles**and the lines

**PQ**and

**QR**touch as shown.

The distance between the

**smallest**and the

**biggest**circles is

16 times the radius of the smallest circle.

What is the size of angle

**PQR**?

simon_geard_diagram.png |

Three circles and the lines PQ and QR touch as shown.The distance between the smallest and the biggest circles is 16 times the radius of the smallest circle. What is the size of angle PQR?
8 Comments
Anand Mehta
10/10/2015 12:24:44 pm
Spoiler : Answer below
Reply
Andrey Dyukmedzhiev
10/10/2015 01:15:05 pm
The distance between the smallest and the biggest circles is the diameter of the red circle, so if we take the blue radius 1, the red radius will be 8, so the distance between the centers of the blue and the red circles is 1 + 8 = 9. The distance between the points of tangency of both circles with the line PQ is 2√(1*8) = 4√2, so
Reply
Simon Geard
11/10/2015 11:15:20 pm
In my answer below I took 'The distance between the smallest and biggest circles' to mean the distance between their centres.
Reply
James McE
10/10/2015 02:11:34 pm
2 * arc sin (7/9) = 1.782.. radians approx
Reply
10/10/2015 04:09:10 pm
102.11 degrees or 1.782 radians
Reply
Simon Geard
11/10/2015 06:48:25 pm
In units of the radius of the smallest circle, let the radii of the other two be a and b, where a < b; let the distance from Q to the first perimeter be c.
Reply
Simon Geard
11/10/2015 09:39:18 pm
I have uploaded an images from a scaled construction:
Reply
Paul
12/10/2015 05:55:34 am
Thank you as always, Guys. Really enjoying watching the discussion evolve. Simon, I've attached your diagram at the bottom of the question, so that people can access it easily. Many thanks for this contribution.
Reply
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