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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?

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

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

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?

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

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 B)) .

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. -_-

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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?

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.

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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.

post-1048-1230871984.gif
What are the six points [u-z]?
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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.

post-1048-1230871984.gif
What are the six points [u-z]?

[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.

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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.

post-1048-1230975647.gifpost-1048-1230975488.gif

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.

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." :)

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