Circles in Circles
A large circle is drawn with a smaller circle sharing the same center. A third circle is drawn so that it is tangent to the small inner circle and the large circle creating two lines tangent to the two smaller circles and intersecting at a point along the larger circle. What is the angle of the intersection of the two tangent lines in degrees?
The answer is 27.309.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Edit Unsubscribe
Identify the radii of the three circles as X,Y,and Z as shown and draw in the radii to create similar triangles for both inner circles.
You can create a proportion for the hypotenuse an side of each triangle resulting in the following proportion: Y X = Z X + Y + Z
After cross multiplying we get XZ = XY + Y^2 + YZ which can be regrouped to XZ - XY = Y^2 + YZ which can be further organized to X(Z - Y) = Y(Y + Z)
Since the radius of the smallest circle and the diameter of the medium circle are equal to the radius of the large circle we get the equation
X = Y + 2Z or solving for Z, Z = 2 X − Y
Substituting in for Z and simplifying gives X ( 2 X − Y ) = Y(Y + 2 X − Y
Expanding this out gives and multiplying by two gives X^2 - XY = 2XY + 2Y^2 + XY - Y^2
Grouping and organizing brings us to the quadratic Y^2 + 4XY - X^2 = 0
Applying the Quadratic Formula results in Y = X (-2 +/- 2.23607) only the positive solution works so the smaller circle has a radius that is 0.23607 of the large circle.
Applying inverse sin to the smaller triangle yields an angle of 13.6545. Doubling that to find the total angle yields 27.309 degrees.