Suppose you're given a circle with center $O$, I'm curious, how can one construct with ruler and compass three circles inside the larger circle such that each is tangent to the larger circle as well as to the other two?
[Math] Three tangent circles inside a larger circle
circlesgeometry
Related Solutions
Let's get a figure in here, eh?
We can exploit the fact that the points of mutual tangency of the three inner circles (red points in the figure) form an equilateral triangle; we also know that the red points are the midpoints of the segments formed by joining any two of the three blue points (the centers of the inner circles).
We then deduce that the triangle formed by the red points is half the scale of of the triangle formed by the blue points, and find that the equilateral triangle formed by the blue points has a side length of 6 ft., and that the inner circles have a radius of 3 ft. Using the law of cosines, we reckon that the distance from a blue point to the center of the triangle formed by the blue points is $2\sqrt{3}$ ft. Adding to that the radius of an inner circle, we find that the radius of the outer circle is $3+2\sqrt{3}$ ft.
The area of the outer circle is $\approx$ 131.27 square feet.
Your two purple circles intersect. Draw a line between the two intersections. This is called the chordal or the radical axis... The center of any circle that is perpendicular to your two given purple circles lies along that line.
page 153 in Dorrie.
Best Answer
In the image above are sequential steps of how I did it (left image is the earlier steps and the right the later steps). I'll only explain the figure constructed in the center. After having divided the disk into three (left figure) by the three radius, construct an arbitrary chord S perpendicular to AB. Construct circle centered at T tangent to the given circle at U. Construct a line passing U and the point at which the perpendicular and the the circle centered at T intersects. The line should intersect at diameter AB which is the tangent. The rest is easier to follow from the figure.
I was looking for the answer for this problem in the net but to no avail. After much trial and error, got this construction. Not 100% sure though, kinda new to this non-coordinate based geometry.