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March 21, 2020

Challenge Problem 8: Solve For The Radius, 2017 AMC 12A Problem 16

As shown in the diagram below, circle D is internally tangent to circle A and externally tangent to circles B and C. If circle B has a radius of 2 and C has a radius of 1, what is the radius of circle D?


I will present one way to solve the problem that involves only high school geometry. There are many several other ways to solve the problem, presented here.

First I will mention an intuitive but important concept. If two circles are tangent, then their centers and the tangent point must all be collinear (on the same line). We can see this visually below. Draw a line through the tangent point C. This line is mutually tangent to each circle’s radius, and so the two radii meet at C at a 180 degree angle (a straight line). Hence all three points are collinear. The statement is true for both externally and internally tangent circles.


We can apply this idea to the problem at hand. Construct line segments between the center’s of circles to form BD and DC. Also construct AP which must be collinear with AD.


Suppose circle D has a radius r. Then BD = 2 + r and CD = 1 + r. Also AD = AP – DP = 3 – r.

The diameter of circle A is twice the sum of the radii of B and C, so the diameter is 2(2+1) = 6. Hence circle A has a radius of 6/2 = 3.

Consequently AB = radius A – radius B = 3 – 2 = 1, and AC = radius of A – radius C = 3 – 1 = 2.

Now let’s focus on the triangles formed by the centers of circles as shown in the following diagram.


Denote angle C by θ. We now apply Al-Kashi’s law of cosines to triangles BDC and ADC to get the following two equations:

(2 + r)^2 = (1 + r)^2 + 3^2 – 2(1 + r)(3)cos θ
(3 – r)^2 = (1 + r)^2 + 2^2 – 2(1 + r)(2)cos θ

Multiplying the second equation by 1.5 and then subtracting from the first eliminates the term with θ, giving:

(2 + r)^2 – 1.5(3 – r)^2 = -0.5(1 + r)^2 + 3^2 – 1.5(2^2)
4 + 4r + r^2 – 13.5 + 9r – 1.5r^2 = -0.5 – r – 0.5r^2 + 9 – 6
14r = 12
r = 6/7

The equation simplifies tremendously and we have solved that circle D has a radius of 6/7.

Source

Thanks to JY from Indonesia for suggesting a similar problem!

AoPS page for problem 16 on 2017 AMC 12A
https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_12A_Problems/Problem_16
Mind Your Decision:
https://mindyourdecisions.com/blog/2020/02/09/solve-for-the-radius-2017-amc-12a-problem-16/

Yêu Tiếng Anh

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