bonanova 84 Posted January 1, 2009 Report Share Posted January 1, 2009 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? Quote Link to post Share on other sites

0 Guest Posted January 1, 2009 Report Share Posted January 1, 2009 (edited) 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 January 1, 2009 by Lost in space Quote Link to post Share on other sites

0 bonanova 84 Posted January 1, 2009 Author Report Share Posted January 1, 2009 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? Quote Link to post Share on other sites

0 Guest Posted January 1, 2009 Report Share Posted January 1, 2009 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 . 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. Quote Link to post Share on other sites

0 Guest Posted January 1, 2009 Report Share Posted January 1, 2009 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. Quote Link to post Share on other sites

0 bonanova 84 Posted January 2, 2009 Author Report Share Posted January 2, 2009 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.What are the six points [u-z]? Quote Link to post Share on other sites

0 Guest Posted January 2, 2009 Report Share Posted January 2, 2009 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.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. Quote Link to post Share on other sites

0 Guest Posted January 2, 2009 Report Share Posted January 2, 2009 (edited) To me it looks like...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 Quote Link to post Share on other sites

0 bonanova 84 Posted January 3, 2009 Author Report Share Posted January 3, 2009 [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. Can you imagine a philosophical thought consistent with these facts and the flavor text? Quote Link to post Share on other sites

0 bonanova 84 Posted January 20, 2009 Author Report Share Posted January 20, 2009 Can you imagine a philosophical thought consistent with these facts and the flavor text?When a person strives for his highest possible attainment, he discovers an ideal that touches him at the center of his being. Quote Link to post Share on other sites

0 Prime 15 Posted January 30, 2009 Report Share Posted January 30, 2009 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." Quote Link to post Share on other sites

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## bonanova 84

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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