Newton
by math_explorer, Apr 15, 2011, 4:49 AM
Google fails me again.
Right, so "Newton's theorem" can mean any of the following.
1. Newton's binomial theorem, which is basically the binomial theorem with the exponent and the binomial's upper parameter generalized to all reals. Or complex numbers, apparently.
2. A quadrilateral with an inscribed circle is given. Draw the diagonals; connect the points of tangency on opposite sides. The four resulting lines concur.
This can be proved through some creative Pascal. It's also the degenerate case of Brianchon.
3. Newton's method for approximating zeroes of a function. Can't imagine that showing up in olympiads.
4. Something about abstract algebraic curves and transversals that I do not even want to try to interpret. The thing on Wikipedia and MathWorld.
5. Something that apparently involves harmonic conjugates. Haven't found this yet but I think it's an easy property.
6. The midpoints of the three diagonals of a complete quadrilateral are collinear. This line, appropriately, is known as the Newton line or Newton-Gauss line.(The three diagonals of
are 
, 
, and the connections of the intersections of 
and 
respectively.)
Cut the Knot has two cute and reasonably natural proof, one with medial segments and a sort of parallelogram net, and one with medial segments and Menelaus.
In fact, the circles with the diagonals as diameters are coaxial. Coaxial circles' centers are collinear (say that five times fast!) Let's go off on a tangent to prove this.
Let
be a triangle circumscribed by 
and 
be its orthocenter.
"Obviously" (read: angle chase) the reflections of about 
, 
, 
all lie on . Now we can take the power of a point on 
, and divide by 2. If 
are the feet of , the result is 
, but since our previous computations were symmetric about 
this is the same for 
and 
. Now if 
then is the power of to the circle with diameter 
. So, has constant power of a point to any circle whose diameter is a cevian of . (Wow!)
Now, the punchline: the sides of a quadrilateral determine four triangles (pick three sides of the quadrilateral and extend until they intersect), whose orthocenters are all different (just try picking a few parallel altitudes), and every diagonal is a cevian of every such triangle, so taking circles with the diagonals as the diameters, you find that they have four different radical centers! This is only possible if they're coaxial. Boom.
Those radical axes sure get around...
Right, so "Newton's theorem" can mean any of the following.
1. Newton's binomial theorem, which is basically the binomial theorem with the exponent and the binomial's upper parameter generalized to all reals. Or complex numbers, apparently.
2. A quadrilateral with an inscribed circle is given. Draw the diagonals; connect the points of tangency on opposite sides. The four resulting lines concur.
This can be proved through some creative Pascal. It's also the degenerate case of Brianchon.
3. Newton's method for approximating zeroes of a function. Can't imagine that showing up in olympiads.
4. Something about abstract algebraic curves and transversals that I do not even want to try to interpret. The thing on Wikipedia and MathWorld.
5. Something that apparently involves harmonic conjugates. Haven't found this yet but I think it's an easy property.
6. The midpoints of the three diagonals of a complete quadrilateral are collinear. This line, appropriately, is known as the Newton line or Newton-Gauss line.(The three diagonals of
are 
, 
, and the connections of the intersections of 
and 
respectively.)Cut the Knot has two cute and reasonably natural proof, one with medial segments and a sort of parallelogram net, and one with medial segments and Menelaus.
In fact, the circles with the diagonals as diameters are coaxial. Coaxial circles' centers are collinear (say that five times fast!) Let's go off on a tangent to prove this.
Let
be a triangle circumscribed by 
and 
be its orthocenter."Obviously" (read: angle chase) the reflections of
about 
, 
, 
all lie on . Now we can take the power of a point on 
, and divide by 2. If 
are the feet of , the result is 
, but since our previous computations were symmetric about 
this is the same for 
and 
. Now if 
then is the power of to the circle with diameter 
. So, has constant power of a point to any circle whose diameter is a cevian of . (Wow!)Now, the punchline: the sides of a quadrilateral determine four triangles (pick three sides of the quadrilateral and extend until they intersect), whose orthocenters are all different (just try picking a few parallel altitudes), and every diagonal is a cevian of every such triangle, so taking circles with the diagonals as the diameters, you find that they have four different radical centers! This is only possible if they're coaxial. Boom.
Those radical axes sure get around...
This post has been edited 2 times. Last edited by math_explorer, Aug 19, 2011, 8:20 AM
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