Centroid: The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2:1.
[This message has been edited by randye (edited 05-28-2012).]
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05:19 PM
lurker Member
Posts: 12355 From: salisbury nc usa Registered: Feb 2002
You draw a circle inside the triangle that touches all 3 sides of the triangle and the center of the circle is the center of the triangle.
Sounds plausible to me, but if you look at Randye's drawing, your center would be way off from his. Which one is right ? we would have to blow it up to larger size, cut it out of cardboard and see if its either or neither. Just by looking at it, with no real knowlege, I would think the center is going to be like his and closer to the largest side to balance it.
[This message has been edited by rogergarrison (edited 05-28-2012).]
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08:17 PM
lurker Member
Posts: 12355 From: salisbury nc usa Registered: Feb 2002
A water treatment plant in Toronto? Does it have something to do with that specific geographical location being suited for balancing objects because of the earths gravity and rotation? Kinda like the make the broom stand up by itself phenomenon except localized.
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09:36 PM
May 29th, 2012
Marvin McInnis Member
Posts: 11599 From: ~ Kansas City, USA Registered: Apr 2002
Centroid: The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2:1.
Correct. Any two of the three sides should give the same result.
quote
Originally posted by Rallaster:
You draw a circle inside the triangle that touches all 3 sides of the triangle and the center of the circle is the center of the triangle.
Nope. That would be correct only for an equilateral triangle (or any other "regular "polygon). What you describe is called the "medial point" ... very useful for many problems from medical treatment to targeting nuclear weapons, but it's not the same as the centroid (center of gravity). For one thing, the centroid is not guaranteed to be within the object (think of a crescent), but the medial point always is.
The centroid and the medial point are guaranteed to be collocated only for convex regular polygons ... i.e. convex polygons with all equal angles and equal sides, Finding the medial point(s) of an arbitrary polygon (e.g. the map of a country, state, or county) is far from trivial. Consider a simple rectangle. It has only one centroid, but it will have an infinite number of medial points falling along what's called the "medial axis." An arbitrary polygon can have one or many medial points, and they may or may not be connected. Think of a symmetrical dumbbell shape, which has two medial points that aren't connected.
(I have just summarized something that once occupied several months of my life.)
[This message has been edited by Marvin McInnis (edited 06-05-2012).]
