Circle amongst circles
Three circles are all touching but not overlapping. All have a radius of 12. A blue equilateral triangle is squeezed in between them and another red circle squeezed between the triangle and circles.
What is the radius of the red circle?
The answer is 1.
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1 solution
Note: All angles are calculated based on the fact that the black outermost triangle is equilateral so its angles are 60 degrees. From there, its just bisection of angles and use of (sum of angles = 180deg in a triangle) to find all necessary angles in the calculative steps below.
let d = distance from circumference to circumcenter of blue triangle
d = (sqrt(24^2 - 12^2) - 12*tan(30) - 12)
let r = radius of red circle
h = height of pink triangle = r + d*sin(30) = r + height from bottom of blue triangle to its circumcenter.
pink triangle lateral line distance = 12+r
pink triangle longest line distance = 12+d
angle = 60
Using cosine rule,
(12+r)^2 = (r+0.5d)^2 + (12+d)^2 - 2(r+0.5d)(12+d)cos(60)
r = (1/48) * d(d+24)
= 1.
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