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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,705

The bear that looked like a rock

It was said that the great explorer Juan Ponce de Leon sailed to the new world with a map of an island that contained the fountain of youth. This island was off the coast of Florida which he named.

The map said that on the island he one would find a rock that looked like a bear. Close by was a bear that looked like a rock. He was instructed to leave the bear alone so he is no longer important in the story.

At some distance east from the rock that looked like a bear was a large maple tree. Also, it was written there was a huge sinkhole far from the rock.

The map instructed him to begin at the rock that looked like a bear, avoid the bear that looked like a rock and walk to the maple tree. Then he would turn right at right angles to his previous direction and walk an equal distance as from the rock that looked like a bear to the maple tree.

The map instructed him to build a small marker there. He then was to walk back to the rock that looked like a bear carefully avoiding the bear that looked like a rock.

Then he was to set out from the rock that looked liked a bear carefully avoiding you know who and travel to the sinkhole. Upon reaching it he was to make a right angle turn left of his previous direction. He would travel an equal distance in that direction as he had just traveled from the rock that looked like a bear to the sinkhole. When he did that he was to construct a second marker on that spot.

Here is the great thing, the fountain of youth was located exactly between the 2 markers!

So Juan set sail with a trusty crew and 251 cans of pepper spray for the bear that looked like a rock. He found Florida, named it like history recorded and found the island! It was much larger than he expected but he found the sinkhole and the only maple tree but there was no rock that looked like a bear?!

The Osceola Indians who inhabited the island hauled the rock away to prevent anyone from discovering the location of the fountain of youth.

Try as he might he could not figure where the fountain was located and he sailed away in disgust at his failure.

Now Juan was a great explorer but he did not have Geogebra. We do!

Let's assume we know the position of the rock that looked like a bear and see what we can discover.

1) Create a point on the screen and label it rock.

2) Create another point to the right of the point named rock and call it maple.

3) Create a third point to the right of the rock and below the point called maple. Call it Sinkhole.

The position chosen for the three points is just for convenience.

4) Draw a line segment from rock to maple. It will be labelled a.

5) Draw a cirlcle with radius = a and using the point maple as the center.

6) Now draw a line that is perpendicular to "a" through the point called maple. It will be called c.

7) Find the intersection of c and the circle using the intersection tool. 2 points will be created so use the one the that is to the right of a. The intersection will be called A. Rename it to Marker1.

8) Hide the circle, c and a.

9) Draw a line segment from the rock to the sinkhole. It will be called d.

10) Draw a perpendicular line to d through sinkhole. called e.

11) Draw a circle through sinkhole with radius = d then get the intersection of that circle with the perpendicular e.

12) Choose the point to the left of d it will be called D and rename it to Marker2.

13) Hide all objects other than the 5 points rock, sinkhole, maple, Marker1, Marker2.

14) Draw a line segment between Marker1 and Marker2 and get the midpoint of that line which on mine is called A. Hide the line segment and rename A to FOY.

Now take and move the point called rock and be careful of the bear that looks like a rock. Notice that no matter where you move it the position of the FOY never changes! That means Juan did not need the position of the rock that looks like a bear. The maple tree and the sinkhole were enough.

Check your drawing against mine.

If only Juan Ponce de Leon had had Geogebra...

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

So, I was right.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,705

I am not following you.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Have you seen the PM?

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,705

Yes, I have seen it. What about it?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Doesn't matter.

Where did you get this problem?

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,705

Two sources provide the problem.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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