Math Is Fun Forum

jeff787

Member

Registered: 2015-04-08

Posts: 7

R. Blum equation of tangents

In some ways this question is an extension of:

http://mathisfunforum.com/viewtopic.php?id=14945

My question arises from an exercise in Coxeter's Intro to Geometry,
pg 114 ex4. He references Richard Blum's paper (1958) that I was able
to track down in books.google.com.

On page 2 of the paper, Blum gives an equation of the tangents that I
have not been able to make sense of. I did derive the equation of the
tangent circle in his oblique coordinates. However the next step he gives
eludes me, i.e. the C(p,n)*C(p0,n0) - C^2(p,n | p0,n0) equation for the
tangent lines.

Although I could beat my head against the wall a bit longer, I figure it's
time to move. So I would appreciate any pointers to making sense of it
that anyone could provide. Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: R. Blum equation of tangents

hi jeff787

Welcome to the forum.

The diagram from that thread is not showing for me.  The forum was updated after that date and the old method of image upload no longer works.  Are you able to post a fresh diagram?

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

jeff787

Member

Registered: 2015-04-08

Posts: 7

Re: R. Blum equation of tangents

Hi Bob,

Regarding the old thread, I'm not sure that it will be necessary.
In any case, I'm not sure what you want me to post since there
are possibly many graphics scattered through the thread. (Odd
though that it displays okay for me on my tablet.)

Anyway, the pertinent animal to look at is:

https://books.google.com/books?id=N1t02 … CE8Q6AEwBw

Again, the confusing aspect is on page 2 regarding the equation of the tangent lines.
One confusion is whether that equation is a starting point that he
pulled from another source or a finishing point that summarizes two
equations of lines in one easy to provide form. The second confusion is
the exact form of the "polar form" term.

-jeff

Last edited by jeff787 (2015-04-08 06:21:16)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: R. Blum equation of tangents

hi Jeff,

Thanks for the link.  Hhmmm.  He didn't make it easy did he?  The diagram isn't next to the text so you have to keep flitting back and forth, many points aren't labelled and there's no triangle shown.  So I'm really confused about this and don't know if I'll ever be able to follow what you want, let alone help with an answer.

It would help if you could reproduce the diagram with some letters to identify what is what, and then maybe I can follow what you are seeking.

Thanks,

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

jeff787

Member

Registered: 2015-04-08

Posts: 7

Re: R. Blum equation of tangents

Thanks for the link.  Hhmmm.  He didn't make it easy did he?  The diagram isn't next to the text so you have to keep flitting back and forth, many points aren't labelled and there's no triangle shown.  So I'm really confused about this and don't know if I'll ever be able to follow what you want, let alone help with an answer.

Nope. Not easy. Not easy at all. His target audience was professional mathematicians. I'm not sure whether he even made it too confusing for them. Coxeter didn't complain about the article in his text. Like you it was confusing for me too initially. However I did make it as far as the equation for tangents before getting stuck. I think I would put the problem at either a fourth year undergraduate or first year graduate student level. I can certainly help out in clearing up confusion up to the point of the equation of tangents.

It would help if you could reproduce the diagram with some letters to identify what is what, and then maybe I can follow what you are seeking.

Hmmm, we may have to make do with pencil and paper and maybe a compass and straightedge here. I don't think I have it in me to get together the software tools (and familiarity with them) to put together graphics that in the end just mimic what was there in the paper. If you do have them convenient and want to make a stab at the figure then we could definitely use that. However in its absence we can use the diagram in the paper, either reproduced by hand or captured via a screen shot, for easy reference.

To possibly help with clearing up some of the confusion, let me give a little bit of overview. The xi and eta coordinates that he provides are tied to one of the common tangents of the 9 point circle and of any one of the excircles or the incircle, *and* a side of the triangle tangent to whichever incircle or excircle was chosen. His diagram to me seems to say that he went towards the incircle, he doesn't say. Anyway, a (somewhat more understandable) diagram can then be created by drawing two circles tangent to each other, making the tangent one of the axises(sp?) and filling in a triangle for that 9pt circle and tangent circle and using one of the sides of the triangle that intersect with the common tangent (now an axis) as the other axis. In his diagram eta references the common tangent and xi is one of sides of the triangle that intersects the eta axis. *Note* the two axises in all probability are not orthogonal. For this particular diagram, that is unavoidable.

At this point one then works his way through the sequence of key equations in the paper...until of course one gets stuck and goes begging for help. In any case I hope the above helps in clearing up a bit of the confusion.

-jeff

Last edited by jeff787 (2015-04-09 03:29:46)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: R. Blum equation of tangents

hi Jeff,

Maybe this will help.

The triangle is ABC,

D,E and F are the midpoints of the sides and the three red lines are the perpendicular bisectors leading to the circumcentre. (not labelled and circle left out for clarity)

G, H and I are the 'feet' of the perpendiculars from the vertices to the opposite sides and N is the orthocentre. (circle left out)

J, K and L are the midpoints of AN, BN and CN respectively.

M is the centre of the circle that goes through these nine points.

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

jeff787

Member

Registered: 2015-04-08

Posts: 7

Re: R. Blum equation of tangents

Okay. That illustration has the 9 point circle, but what of the circle that is tangent to it, i.e. one of the excircles or the incircle?

If needbe there is a procedure for recreating a triangle from for example the 9 point circle and a tangent excircle, but I don't think it's really necessary for setting up the illustration. All that is necessary in your illustration is to add one of those circles and then add the two oblique axises. (I would suggest using the incircle to make the illustration look somewhat like Blum's figure 1.) Once all of necessary elements are in the illustration, the extraneous items (extraneous with respect to the problem) might be deleted for a cleaner figure. However, that's more up to you as to what you want the figure to represent.

Notice how his figure two is with an excircle.

Does the above help?

-jeff

Last edited by jeff787 (2015-04-09 10:15:33)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: R. Blum equation of tangents

M is the centre of the 9 point circle.  (green)
O is the centre of the inscribed circle.  (red)
P is the point common to both (ie. where they meet tangentially)

Now what?

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob