In english this time: take a triangle (call the points p,q,r). Draw a perpendicular line from each line to its opposite point (say going from line [qr] to point p, and so on). These intersect at the centroid of the triangle (known geometric theorem).
A line has the equation ax + by + c = 0, and its slope (m) can be expressed as -b/a. The line perpendicular to this has the slope a/b (negative reciprocal refered to in the story).
What the new theorem says is: make the new slope -a/b (the non-negative reciprocal). And find one line using each slope. There's a new point, heretofore undiscovered, at their intersection.
What the story doesn't say is what point to use. Every line needs a point and a slope to find its equation. Do we use the original points (p,q,r) and the new slopes, or the new slopes and the points at the intersections of the old perpendicular lines and the original lines of the triangle. The maa site is slashdotted, so I can't find out either
My second question is: Why doesnt everybody understand this? It does only take 8th grade maths to get, maybe 10th grade if you weren't in the advanced class. I know that nobody where I'm from (shameless plug: Ireland) can get into college without knowing this, because several questions of the exams that everybody takes are based on this. It may take a genius to discover this (I question the use of the word genius; I'm putting it down to boredom and curiousity), but everybody above the age of 15 should get this, and you 13 year old 133t h4x0rs who read this, this should be well within your comprehension.
OK take a triangle (call the points p,q,r). Draw a perpendicular line from each line to its opposite point (say going from line [qr] to point p, and so on). These intersect at the centroid of the triangle (known geometric theorem).
A line has the equation ax + by + c = 0, and its slope (m) can be expressed as -b/a. The line perpendicular to this has the slope a/b (negative reciprocal refered to in the story).
What the new theorem says is: make the new slope -a/b (the non-negative reciprocal). And find one line using each slope. There's a new point, heretofore undiscovered, at their intersection.
What the story doesn't say is what point to use. Every line needs a point and a slope to find its equation. Do we use the original points (p,q,r) and the new slopes, or the new slopes and the points at the intersections of the old perpendicular lines and the original lines of the triangle
My second question is: Why doesnt everybody understand this? It does only take 8th grade maths to get, maybe 10th grade if you weren't in the advanced class. I know that nobody where I'm from (shameless plug: Ireland) can get into college without knowing this, because several questions of the exams that everybody takes are based on this. It may take a genius to discover this (I question the use of the word genius; I'm putting it down to boredom and curiousity), but everybody above the age of 15 should get this, and you 13 year old 133t h4x0rs who read this, this should be well within your comprehension.
You're confusing theorems and theorys. Theorems are geometric laws, and can be proven. Theorys can also be proven, but then they're not theorys anymore, they become laws:) Take the most simple geometric theorem, that the external angle at a point on a triangle is equal to the sum the opposite angles of that triangle:
It is known that the sum of the internal angles of a triangle is 180 degrees
It is known that a straight line (sum of internal and external angles) is 180 degrees Take out the common angle, and you're left with the sum of the internal angles, an equals sign, and the external angle Proven
Slashdot Mirror
In english this time: take a triangle (call the points p,q,r). Draw a perpendicular line from each line to its opposite point (say going from line [qr] to point p, and so on). These intersect at the centroid of the triangle (known geometric theorem).
A line has the equation ax + by + c = 0, and its slope (m) can be expressed as -b/a. The line perpendicular to this has the slope a/b (negative reciprocal refered to in the story).
What the new theorem says is: make the new slope -a/b (the non-negative reciprocal). And find one line using each slope. There's a new point, heretofore undiscovered, at their intersection.
What the story doesn't say is what point to use. Every line needs a point and a slope to find its equation. Do we use the original points (p,q,r) and the new slopes, or the new slopes and the points at the intersections of the old perpendicular lines and the original lines of the triangle. The maa site is slashdotted, so I can't find out either
My second question is: Why doesnt everybody understand this? It does only take 8th grade maths to get, maybe 10th grade if you weren't in the advanced class. I know that nobody where I'm from (shameless plug: Ireland) can get into college without knowing this, because several questions of the exams that everybody takes are based on this. It may take a genius to discover this (I question the use of the word genius; I'm putting it down to boredom and curiousity), but everybody above the age of 15 should get this, and you 13 year old 133t h4x0rs who read this, this should be well within your comprehension.
OK take a triangle (call the points p,q,r). Draw a perpendicular line from each line to its opposite point (say going from line [qr] to point p, and so on). These intersect at the centroid of the triangle (known geometric theorem).
A line has the equation ax + by + c = 0, and its slope (m) can be expressed as -b/a. The line perpendicular to this has the slope a/b (negative reciprocal refered to in the story).
What the new theorem says is: make the new slope -a/b (the non-negative reciprocal). And find one line using each slope. There's a new point, heretofore undiscovered, at their intersection.
What the story doesn't say is what point to use. Every line needs a point and a slope to find its equation. Do we use the original points (p,q,r) and the new slopes, or the new slopes and the points at the intersections of the old perpendicular lines and the original lines of the triangle
My second question is: Why doesnt everybody understand this? It does only take 8th grade maths to get, maybe 10th grade if you weren't in the advanced class. I know that nobody where I'm from (shameless plug: Ireland) can get into college without knowing this, because several questions of the exams that everybody takes are based on this. It may take a genius to discover this (I question the use of the word genius; I'm putting it down to boredom and curiousity), but everybody above the age of 15 should get this, and you 13 year old 133t h4x0rs who read this, this should be well within your comprehension.
- It is known that the sum of the internal angles of a triangle is 180 degrees
See, you can prove theoremsIt is known that a straight line (sum of internal and external angles) is 180 degrees
Take out the common angle, and you're left with the sum of the internal angles, an equals sign, and the external angle
Proven