With Thanks to James Taylor

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With Thanks to James Taylor
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With Thanks to James TaylorMusic is true. An
octave is a mathematical reality. So is a 5th.
So is a major 7th chord.And I have the feeling
that these have emotional meanings to us,not only
because we're taught that a major 7th is warm and
fuzzy and a diminished is sort of threatening and
dark,but also because they actually do have these
meanings. It's almost like it's a language
that's not a matter of our choosing.It's a
truth. The laws of physics apply to music, and
music follows that.So it really lifts us out of
this subjective, opinionated human position and
drops us into the cosmic picture just like that.
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With Thanks to James TaylorMusic is true. An
octave is a mathematical reality. So is a 5th.
So is a major 7th chord.And I have the feeling
that these have emotional meanings to us,not only
because we're taught that a major 7th is warm and
fuzzy and a diminished is sort of threatening and
dark,but also because they actually do have these
meanings. It's almost like it's a language
that's not a matter of our choosing.It's a
truth. The laws of physics apply to music, and
music follows that.So it really lifts us out of
this subjective, opinionated human position and
drops us into the cosmic picture just like that.
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Music is true. An octave is a mathematical
reality. So is a 5th. So is a major 7th
chord.And I have the feeling that these have
emotional meanings to us,not only because we're
taught that a major 7th is warm and fuzzy and a
diminished is sort of threatening and dark,but
also because they actually do have these
meanings. It's almost like it's a language
that's not a matter of our choosing.It's a
truth. The laws of physics apply to music, and
music follows that.So it really lifts us out of
this subjective, opinionated human position and
drops us into the cosmic picture just like
that.-- recording artist James Taylor, in the
May 2002 Performing Songwriter
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Problem 1
Examine the orthocentre of the triangle which has
its 3 vertices on the two branches of a
rectangular hyperbola.
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Problem 2
P lies on a hyperbola with focus S. The
perpendicular from the focus S onto the tangent
meets it at T.Show that T lies on the auxilliary
circle.
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Problem 3

• Prove that tan a tan b tan g tan a ? tan b
  ? tan g
• if a, b and g are the angles of a triangle

a
b
g
sketchpad
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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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Proof

• tan a tan b tan g tan a ? tan b ? tan g

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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

Now remember arithmetic and geometric means.
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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

Now remember arithmetic and geometric means.
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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

Now remember arithmetic and geometric means.
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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

Now remember arithmetic and geometric means.
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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

Now remember arithmetic and geometric means.
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An interesting consequence

• tan a tan b tan g tan a ? tan b ? tan g

So in any triangle the product (or sum) of the
tangents of the three angles is never less than
3?3
With equality if the triangle is equilateral!
sketchpad
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The Tangent Rule

• This rule has all but disappeared from modern
  text books but is still a nice piece of
  mathematics.
• The tangent rule says that

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The Tangent Rule

• This rule has all but disappeared from modern
  text books but is still a nice piece of
  mathematics.
• The tangent rule says that

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The Tangent Rule the un-proof

• (not recommended)

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The Tangent Rule the un-proof

• (not recommended)

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The Tangent Rule the un-proof

• (not recommended)

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The Tangent Rule the un-proof

• (not recommended)

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The Tangent Rule the proof

• (recommended)

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The Tangent Rule the proof

• (recommended)

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The Tangent Rule the proof

• (recommended)

and using sums and differences
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The Tangent Rule the proof

• (recommended)

and using sums and differences
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The Tangent Rule the proof

• (recommended)

and using sums and differences
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Problem 4
Arithmetic and Geometric Mean
Totally illegal!!
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Problem 4
Arithmetic and Geometric Mean
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Problem 4
Arithmetic and Geometric Mean
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Problem 4
Arithmetic and Geometric Mean
Now write it backwards
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Problem 4
Arithmetic and Geometric Mean
Conceal the crime!!
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Problem 5
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Problem 5
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Problem 6
Prove that the area of a triangle never exceeds
one sixth the sum of the squares of the lengths
of the sides
Let ? be the area of the triangle ABC
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Problem 6
Prove that the area of a triangle never exceeds
one sixth the sum of the squares of the lengths
of the sides
Let ? be the area of the triangle ABC
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Problem 6
Prove that the area of a triangle never exceeds
one sixth the sum of the squares of the lengths
of the sides
Let ? be the area of the triangle ABC
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Problem 6
Prove that the area of a triangle never exceeds
one sixth the sum of the squares of the lengths
of the sides
Let ? be the area of the triangle ABC
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Problem 6
Prove that the area of a triangle never exceeds
one sixth the sum of the squares of the lengths
of the sides
Let ? be the area of the triangle ABC
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Problem 7
An amazing sequence
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Problem 7
An amazing sequence
show that an and bn converge to the same limit
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Problem 7
An amazing sequence
show that an and bn converge to the same limit
and show that limit is