The Curious Incident of the Dog in the Night-Time - Mark Haddon [77]
This can also be shown by means of the following graph (but this doesn't prove anything):
According to Pythagoras's theorem, if the sum of the squares of the two shorter sides equals the square of the hypotenuse, then the triangle is right-angled. Therefore to prove that the triangle is right-angled we need to show that this is the case.
The sum of the squares of the shorter two sides is (n2 - 1)2 + (2n)2
(n2 - 1)2 + (2n)2 = n4 - 2n2 + 1 + 4n2 = n4 + 2n2 + 1
The square of the hypotenuse is (n2 + 1)2
(n2 + 1)2 = n4 + 2n2 + 1
Therefore the sum of the squares of the shorter two sides is equal to the square of the hypotenuse and the triangle is right-angled.
And the converse of “A triangle with sides that can be written in the form n2 + 1, n2 - 1 and 2n (where n > 1) is right-angled” is “A triangle that is right-angled has sides whose lengths can be written in the form n2 + 1, n2 - 1 and 2n (where n > 1).”
And a counterexample means finding a triangle which is right-angled but whose sides cannot be written in the form n2 + 1,
n2 - 1 and 2n (where n > 1).
So let the hypotenuse of the right-angled triangle ABC be AB.
and let AB = 65
and let BC = 60
Then CA = √ (AB2 - BC2)
= √ (652 - 602) = √ (4225 - 3600) = √ 625 = 25
Let AB = n2 + 1 = 65
then n = √ (65 - 1) = √ 64 = 8
therefore (n2 - 1) = 64 - 1 = 63 ≠ BC = 60 ≠ CA = 25
and 2n = 16 ≠ BC = 60 ≠ CA = 25
Therefore the triangle ABC is right-angled but it does not have sides which can be written in the form n2 + 1, n2 - 1 and 2n (where n > 1). QED
I found this in a book when Mother took me into the library in town in 1996.
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This is not a metaphor, it is a simile, which means that it really did look like there were two very small mice hiding in his nostrils, and if you make a picture in your head of a man with two very small mice hiding in his nostrils, you will know what the police inspector looked like. And a simile is not a lie, unless it is a bad simile.
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But I wouldn't have Shreddies and tea because they are both brown.
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Once I didn't talk to anyone for 5 weeks.
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When I was 6 Mother used to get me to drink strawberry-flavored slimming meals out of a measuring jug and we would have competitions to see how fast I could drink a quarter of a liter.
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People say that you always have to tell the truth. But they do not mean this because you are not allowed to tell old people that they are old and you are not allowed to tell people if they smell funny or if a grown-up has made a fart. And you are not allowed to say “I don't like you” unless that person has been horrible to you.
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Stupid things are things like emptying a jar of peanut butter onto the table in the kitchen and making it level with a knife so it covers all the table right to the edges, or burning things on the gas stove to see what happened to them, like my shoes or silver foil or sugar.
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I only did this once by borrowing the keys when she went into town on the bus, and I hadn't driven a car before and I was 8 years old and 5 months so I drove it into the wall, and the car isn't there anymore because Mother is dead.
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It is permitted to move the chairs and the table in the kitchen because that is different, but it makes me feel dizzy and sick if someone has moved the sofa and the chairs around in the living room or the dining room. Mother used to do this when she did the hoovering, so I made a special plan of where all the furniture was meant to be and did measurements and I put everything back in its proper place afterward and then I felt better. But since Mother died Father hasn't done any hoovering, so that is OK. And Mrs. Shears did the hoovering once but I did groaning and she shouted at Father and she never did it again.
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This is really true because I asked Siobhan