The Golden Ratio is calculated to be 1.618::1.

It describes a pattern that is found in sunflowers, playing cards, architecture and web design layout.

Here are two presentations of a Golden Rectangle where the sides are in Golden Ratio proportion:

Attachment 122576

Now it can be shown that the Golden Ratio is integral to the pentagon, where it is Diagonal Length::Edge Length:

It is therefore possible to construct the blue Golden Rectangle from this knowledge, as shown.

In this Challenge, I wish you to uncover a simpler construction of the Golden Rectangle.

Acorn

P.S. Phi (ɸ) is (√5 + 1) / 2 ≃ 1.6180339887498948482045868343656

Can you Xara-construct a circle whose edge touches three arbitrary points?

The points cannot all lie on the same straight line or create a circle that exceeds Xara's largest dimensions.

Here is a benign example:

You points can instead be the the corners of any triangle.

Acorn

Can you accurately, without calculation, construct a circle from another so the new circle's area is three times larger?

At present I have two construction methods but both involve some in-between scaffolding.

Good thinking,

Acorn

[ATTACH]122537[/ATTACH

Acorn

The key feature is if you remove any one ring to find the other two are not linked.

The follow-up question is how make cuts and re-welds would you need to join three solid rings into a Boromean Ring?

I recall Egg described a method a time back.

Acorn

Again, I stress, only circles can be used in your construction - no math / maths expected, required or wanted.

I chose 100, 200 & 300px diameters but I am happy for any solution where each value is different and, importantly, the solution is independent of the presented sizes.

When I checked for accuracy, I found a gap of 1/100th of a pixel.

Acorn

My design began with a 16cm circle. The next size down was 8cm and the last size is 2cm.

I finally included black edging of thickness 2mm.

Acorn

Again, the construction is to be as simple and as accurate as the Xara Desktop application allows.

This time, any of the circles can be cloned for use in your construction.

You can assume the solution of the previous two Circle Challenges or start afresh.

Acorn

You, however, can create a Red, Green and Blue circle from afresh, each with a diameter of 200px.

The Challenge is to simply, and accurately, position the circles so that each circle equally intersects (overlaps) the other two circles but there is no overlap with all three circles.

Acorn

The Challenge is to use no other shapes. Snapping is allowed.

Acorn