[bonanova]

In Edward Abbott's charming book Flatland, women are needles [2-sided polygons]

and men have 3 or more sides, whose number increased with nobility.

Not surprisingly, circles are priests [infinite nobility].

Clearly the book was written before the age of political correctness.

Although the circular priest differs greatly from the lowly triangular man,

a simple geometric construction shows they nevertheless have six degrees:

not of separation, but of commonality:

Draw an acute [but not regular] triangle and the three lines that represent its altitude.

That means using each of the sides, in turn, as the base.

The points [a, b, c] where bases and altitudes intersect define a circle.

The circle intersects each of the six lines:

the sides (again), at [u, v, w], and the altitudes, at [x, y, z].

What is special about the six points [u, v, w; x, y, z] the priest and man have in common?

[Guest]

You have a way of setting/hiding the problem in a wonderful story

  Reveal hidden contents

find that the centre of the circle has a reference to the triangle or the extentions of each side are related ??? AB+x AC and CB ditto ???

Or are you just trying to say that women are two dimensional - but still pretty creatures of note

edit: do not know the book - btw

Edited January 1, 2009 by Lost in space

[bonanova]

  Lost in space said:

You have a way of setting/hiding the problem in a wonderful story

  Reveal hidden contents

find that the centre of the circle has a reference to the triangle or the extentions of each side are related ??? AB+x AC and CB ditto ???

Or are you just trying to say that women are two dimensional - but still pretty creatures of note

edit: do not know the book - btw

What is/are notable about the six points u,v,w,x,y,z?

[Guest]

  bonanova said:

What is special about the six points [u, v, w; x, y, z] the priest and man have in common?

  Reveal hidden contents

My sketch of the problem was sloppy and inaccurate, but I believe you can use those points, in combination, to create a mirror image of the original triangle.

Again, it's early on New Year's Day and I don't have the tools to draw it properly, so a rough guess is what you'll get for now .

In fact, now that I look at the wording of the OP, I doubt that's even what you're looking for. I need an aspirin.

[Guest]

  bonanova said:

In Edward Abbott's charming book Flatland, women are needles [2-sided polygons]

and men have 3 or more sides, whose number increased with nobility.

Not surprisingly, circles are priests [infinite nobility].

Clearly the book was written before the age of political correctness.

Although the circular priest differs greatly from the lowly triangular man,

a simple geometric construction shows they nevertheless have six degrees:

not of separation, but of commonality:

Draw an acute [but not regular] triangle and the three lines that represent its altitude.

That means using each of the sides, in turn, as the base.

The points [a, b, c] where bases and altitudes intersect define a circle.

The circle intersects each of the six lines:

the sides (again), at [u, v, w], and the altitudes, at [x, y, z].

What is special about the six points [u, v, w; x, y, z] the priest and man have in common?

  Reveal hidden contents

Each set of three points, a,b,c and x,y,z both lie on a circle and define the corners of a triangle that is proportional to the original triangle.

So, within these points the priest 'contains' the triangle, and vice versa.

[bonanova]

  xucam said:

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Each set of three points, a,b,c and x,y,z both lie on a circle and define the corners of a triangle that is proportional to the original triangle.

So, within these points the priest 'contains' the triangle, and vice versa.

Nicely put.

Yes, actually we know that all 9 points a, b, c, u, v, w, x, y, z lie on a circle.

We used a, b, c to find the circle.

The circle then found six other points related to the triangle.

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What are the six points [u-z]?

[Guest]

  bonanova said:

Nicely put.

Yes, actually we know that all 9 points a, b, c, u, v, w, x, y, z lie on a circle.

We used a, b, c to find the circle.

The circle then found six other points related to the triangle.

  Reveal hidden contents

What are the six points [u-z]?

  Reveal hidden contents

[u,v,w] look like the mid points of the sides of the triangle. and [x,y,z] the mid points of the lines AO BO and CO. I'll leave proof as an exercise for the reader.

[Guest]

To me it looks like...

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that the six points [u-Z] mark chords (if thats the correct term) and arcs inside of the circle...

Edited January 2, 2009 by Newt24

[bonanova]

  armcie said:

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[u,v,w] look like the mid points of the sides of the triangle. and [x,y,z] the mid points of the lines AO BO and CO. I'll leave proof as an exercise for the reader.

Nice!

BTW, the triangle does not have to be acute, just neater that way.

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Can you imagine a philosophical thought consistent with these facts and the flavor text?

[bonanova]

  bonanova said:

Can you imagine a philosophical thought consistent with these facts and the flavor text?

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When a person strives for his highest possible attainment, he discovers an ideal that touches him at the center of his being.

[Prime]

  bonanova said:

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When a person strives for his highest possible attainment, he discovers an ideal that touches him at the center of his being.

Very profound! On a lighter and largely unrelated note, HPLD-s come to mind. (Highest Possible Level of Development beings.) They can be found in the "Cyberiad" -- a collection of short stories by Stanislaw Lem. Specifically, in the one story closer to the end entitled "Altruizine Or a True Account of How Bonhomius the Hermetic Hermit Tried to Bring About Universal Happiness, and What Came of It."