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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,946

In triangle ABC, D, E and F are the midpoints of the sides, G, I and H are the feet of the altitudes, N is the orthocentre, and J, K and L are the midpoints of AN, BN and CN.

Theorem.

There is a circle that goes through all of the points D, E, F; G, H, I; J, K, L.

Proof.

Construct the circumcircle of DEF. (I have shown the centre, M, but left out the construction lines to avoid cluttering the diagram.)

As F and D are the midpoints of AB and BC, FD//AC.

As F and J are the midpoints of AB and AN, FJ//BH

But AHB is 90. => JFD is also 90.

We already know that D and F lie on a circle; now we know that J also lies on that circle as JD subtends an angle of 90 at the circumference of the circle. (In fact JD is a diameter.) This follows from one of the angle properties of a circle which you will find here: http://www.mathisfunforum.com/viewtopic.php?id=17799

Similarly K and L will lie on this circle.

JGD is 90. => G is also on this circle.

Similarly H and I will lie on this circle.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,030

Hej Bob!

These threads are a fantastic idea! Nice work.

An intersting fact about the Nine Point Circle (I also know it under the name of Euler's Circle) is that when you apply inversion around the Nine Point Circle, the circumcircle gets mapped into the incircle and vice versa!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,946

Hej Stefy,

Glad you like them. Would you like to add your inversion post ?

I probably need to add the inscribed circle to my list.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,030

Hej Bob!

While I'd love to do that, I don't think I'll have time in the near future.

Oh and yes, inscribed circles are a must!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,946

Hej Stefy,

OK. I'll do inscribed (and maybe exscribed* too) and start to learn about inversion.

Bob

*My spell checker says that's wrong but all my books spell it that way. Maybe 'escribed' ?

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,030

Well, "ex-" sure sounds much better there. "e-" does not seem like a productive suffix.

And besides, you cannot expect a suffix to recognize mathematical terminology, can you? I doubt it would recpgnize "homomorphism" and the like. xD

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,946

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,030

Haha, does it work for all other "common" morphisms? (t.ex. homeomorphism, epimorphism, monomorphism, osv.)

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,946

MS word only recognises homeomorphism from that list. It doesn't like Haha. Nor osv.

But they are all in dictionary.com

Bob

ps. The inscribed circle page is done.

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,030

Well, t.ex. and osv. are abbreviations from Swedish, not English, so that is not too surprising.

I shall take a look at that page.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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