### Post by Don Barone on Apr 28, 2017 18:14:08 GMT -5

Hi Charlotte et al ...

Well it appears that once again the neurons in my brain have become agitated and some new ides are occurring to me. Two days ago I chanced to re-read an old thread where Gary Osborn, a once friend of mind, had suggested some ideas concerning a circle and a square. I had never agreed with his ideas but from his diagram I chanced to discover an ingenious way of finding a 3, 4 and 5 sided right angled triangle using only a circle with diameter 5 units and the sqaure that encloses it and I will give you the diagrams below.

First we start with the circle.

Then we enclose it in a square ...

Now the original problem I proposed was what was the length of the side of the square whose chord length was the same length as it's height off the ground as illustrated in the diagram below.

I strongly suggested that it had to be 4 and thus beautifully illustrated the 3, 4 and 5 sided right angled triangle as illustrated in the diagram below ...

Firstly I drew these lines ...

and then substituted these values ...

and there it has sat for over two years ... until last night and I decided to start drawing more lines ... ( I love drawing lines ) and found that this simple diagram yielded more than I had ever even dreamed of. And so I will post last nights work. However before I do that there is one observation, not my own, that a line drawn from the top right corner to the middle of the bottom line in the sqaure MARKS THE POINT OF THE 4 DISTANCE ON THE CIRCLE AS SHOWN IN THE DIAGRAM BELOW !!

NOW AS WE GO ON WITH THIS STUDY PLEASE TRY TO REMEMBER IT'S SIMPLE ROOTS. ONE SINGLE CIRCLE ENCLOSED IN IT'S CONTAINING SQUARE AND ONE DISTANCE ... OF 5 UNITS NOTHING ELSE IS NEEDED (other than a straight edge and a compass) and also please always try to remember that instead of this 3, 4 5 sided right angled triangle we could just as easily use ... The Pyramid of Khafre or P2 ! It is at this point I will take a short break and ask a very simply question. Which do you think came first ? The Circle or The Square ?

Much, much more to come ...

Cheers

Don Barone

.

Well it appears that once again the neurons in my brain have become agitated and some new ides are occurring to me. Two days ago I chanced to re-read an old thread where Gary Osborn, a once friend of mind, had suggested some ideas concerning a circle and a square. I had never agreed with his ideas but from his diagram I chanced to discover an ingenious way of finding a 3, 4 and 5 sided right angled triangle using only a circle with diameter 5 units and the sqaure that encloses it and I will give you the diagrams below.

First we start with the circle.

Then we enclose it in a square ...

Now the original problem I proposed was what was the length of the side of the square whose chord length was the same length as it's height off the ground as illustrated in the diagram below.

I strongly suggested that it had to be 4 and thus beautifully illustrated the 3, 4 and 5 sided right angled triangle as illustrated in the diagram below ...

Firstly I drew these lines ...

and then substituted these values ...

and there it has sat for over two years ... until last night and I decided to start drawing more lines ... ( I love drawing lines ) and found that this simple diagram yielded more than I had ever even dreamed of. And so I will post last nights work. However before I do that there is one observation, not my own, that a line drawn from the top right corner to the middle of the bottom line in the sqaure MARKS THE POINT OF THE 4 DISTANCE ON THE CIRCLE AS SHOWN IN THE DIAGRAM BELOW !!

NOW AS WE GO ON WITH THIS STUDY PLEASE TRY TO REMEMBER IT'S SIMPLE ROOTS. ONE SINGLE CIRCLE ENCLOSED IN IT'S CONTAINING SQUARE AND ONE DISTANCE ... OF 5 UNITS NOTHING ELSE IS NEEDED (other than a straight edge and a compass) and also please always try to remember that instead of this 3, 4 5 sided right angled triangle we could just as easily use ... The Pyramid of Khafre or P2 ! It is at this point I will take a short break and ask a very simply question. Which do you think came first ? The Circle or The Square ?

Much, much more to come ...

Cheers

Don Barone

.