Question

Question #3e595

Answer 1

Use the standard Euclidean construction tools.

Explanation:

I assume you are either given or can construct the circles with the given radii.

An essential tool is the ability to construct a perpendicular ("right") bisector of a line segment.

Given a line segment `PQ`
with center `P` and radius `abs(PQ)` draw a `color(red)("circle")`
with center `Q` and radius `abs(PQ)` draw a second `color(blue)("circle")`
* connect the two points where the circles intersect to form the right bisector.

For any of the given circles:
draw any two distinct cords `color(red)(AB)` and `color(blue)(CD)`
draw the right bisector for each cord
* the point where the bisector lines intersect will be the center of the circle.

For the given circle (and determined center `X`),
subdivide the circle into 6 equal arcs:
pick an arbitrary point `A` on the circumference of the circle.
with center `A` and circumference `AX` draw another circle
label the points where the new circle intersects the given circle as `B` and `F`
repeat this process using points `B` and `F` as centers (in place of `A`) to establish 2 more points `C` and `E`
* repeat this process one more time with center `C` (or `E`) to establish the 6th point.
The points `A, B, C, D, E, F` divide the circumference of the circle into 6 equal arcs.

Using the arc end points
draw (extended) lines through pairs of points on opposite sides of the given circle; that is through `{A,D}` and through `{B,E}` and through `{C,F}`
draw lines perpendicular to those just constructed through the alternate points `B`, `D`, and `F`
These three new lines will intersect to form an equilateral, circumscribing triangle.