## [EMHL] Re: metric relation in triangle

Expand Messages
• I started from the formulas for R^2, ra^2 and s in terms on a, b, c, namely R^2 = ((a^2 b^2 c^2)/((a + b - c) (a - b + c) (-a + b + c) (a + b + c))), ra^2 =
Message 1 of 7 , Oct 30, 2009
I started from the formulas for R^2, ra^2 and s in terms on a, b, c, namely

R^2 = ((a^2 b^2 c^2)/((a + b - c) (a - b + c) (-a + b + c) (a + b + c))),

ra^2 = -(((a + b - c) (a - b + c) (a + b + c))/(4 (a - b - c)))

s = (a+b+c)/2,

Then I made use of Mathematica to eliminate a,b,c, with the command

Eliminate[{R^2 ==..., ra^2==...,s==(a+b+c)/2,T==ab+bc+ca},{a,b,c}]

Best regards,

Francisco Javier.

--- In Hyacinthos@yahoogroups.com, Kafka Catalin <kafka_mate@...> wrote:
>
> Dear Francisco,
> Thank you very much for information you sent me! When you are available, could you describe me the steps I must follow in order to determine it myself too?!
>
> Best regards,
> Catalin Barbu
> --- On Fri, 10/30/09, Francisco Javier <garciacapitan@...> wrote:
>
> From: Francisco Javier <garciacapitan@...>
> Subject: [EMHL] Re: metric relation in triangle
> To: Hyacinthos@yahoogroups.com
> Date: Friday, October 30, 2009, 12:47 PM
>
>
>
>
>
>
>
>
>
>
>
>
>
> ï¿½
>
>
>
>
>
> Dear Catalin,
>
>
>
> I get the following formula:
>
>
>
> ab + bc + ca =
>
> (s^6 + (4 R ra + 3 ra^2) s^4 + (-16 R^2 ra^2 + 3 ra^4) s^2 -4 R ra^5 + ra^6)/(ra^2 + s^2)^2.
>
>
>
> Best regards,
>
>
>
> Francisco Javier.
>
>
>
> --- In Hyacinthos@yahoogro ups.com, Kafka Catalin <kafka_mate@ ...> wrote:
>
> >
>
> > Dear Hyacinthists,
>
> > Let a,b,c are sides of triangle ABC, s - semiperimeter, R - circumradius and r_a a excircle radius. Find the sum ab+bc+ca in function of s,R,r_a.
>
> > I kow this relation: ab+bc+ca=s^2 + r^2 +4Rr, where r is incircle radius. Exist a relation ab+bc+ca = F(s,R,r_a)?
>
> > Best regards,Catalin Barbu
>
> >
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> [Non-text portions of this message have been removed]
>
• ... The following can all be expressed in terms of R, s, r_a tan(A/2) = r / (s-a) = r_a / s sin(A) = 2 sin(A/2) cos(A/2) = 2 t / (t^2+1), where t=tan(A/2) a =
Message 2 of 7 , Nov 2, 2009
Kafka Catalin <kafka_mate@...> wrote:

> Dear Hyacinthists,
> Let a,b,c are sides of triangle ABC, s - semiperimeter, R - circumradius and r_a a excircle radius. Find the sum ab+bc+ca in function of s,R,r_a.
> I kow this relation: ab+bc+ca=s^2 + r^2 +4Rr, where r is incircle radius. Exist a relation               ab+bc+ca = F(s,R,r_a)?
> Best regards,Catalin Barbu

The following can all be expressed in terms of R, s, r_a

tan(A/2) = r / (s-a) = r_a / s
sin(A) = 2 sin(A/2) cos(A/2) = 2 t / (t^2+1), where t=tan(A/2)
a = 2 R sin(A)
r = (s-a) r_a / s

Then ab+bc+ca = a (2 s - a) + 4 R r s / a
--
Barry Wolk
• Dear Barry, I want obtain  a relation ab+bc+ca in function only s, R and r_a . Sir Francisco Javier send me a relation    ab + bc + ca = (s^6 + (4 R ra +
Message 3 of 7 , Nov 4, 2009
Dear Barry, I want obtain  a relation ab+bc+ca in function only "s, R and r_a". Sir Francisco Javier send me a relation
ab + bc + ca =
(s^6 + (4 R ra + 3 ra^2) s^4 + (-16 R^2 ra^2 + 3 ra^4) s^2 -4 R ra^5 + ra^6)/(ra^2 + s^2)^2 (with Mathematica program), i need a proof a this relation.Best regards, Catalin Barbu
--- On Tue, 11/3/09, wolkbarry <wolkbarry@...> wrote:

From: wolkbarry <wolkbarry@...>
Subject: [EMHL] Re: metric relation in triangle
To: Hyacinthos@yahoogroups.com
Date: Tuesday, November 3, 2009, 2:59 AM

Kafka Catalin <kafka_mate@ ...> wrote:

> Dear Hyacinthists,

> Let a,b,c are sides of triangle ABC, s - semiperimeter, R - circumradius and r_a a excircle radius. Find the sum ab+bc+ca in function of s,R,r_a.

> I kow this relation: ab+bc+ca=s^2 + r^2 +4Rr, where r is incircle radius. Exist a relation               ab+bc+ca = F(s,R,r_a)?

> Best regards,Catalin Barbu

The following can all be expressed in terms of R, s, r_a

tan(A/2) = r / (s-a) = r_a / s

sin(A) = 2 sin(A/2) cos(A/2) = 2 t / (t^2+1), where t=tan(A/2)

a = 2 R sin(A)

r = (s-a) r_a / s

Then ab+bc+ca = a (2 s - a) + 4 R r s / a

--

Barry Wolk

[Non-text portions of this message have been removed]
• Dear Catalin, ... Barry sent you the following: If t = tan(A/2) = ra/s then a = 2.R.sinA = 4.R.t/(1 + t^2) = 4.R.s.ra/(s^2 + ra^2) (1) bc = (abc)/a = 2.R.S/a
Message 4 of 7 , Nov 4, 2009
Dear Catalin,

> Dear Barry, I want obtain  a relation ab+bc+ca in
> function only "s, R and r_a". Sir Francisco Javier send me a
> relation
> ab + bc + ca =
> (s^6 + (4 R ra + 3 ra^2) s^4 + (-16 R^2 ra^2 + 3 ra^4) s^2
> -4 R ra^5 + ra^6)/(ra^2 + s^2)^2 (with Mathematica program),
> i need a proof a this relation.

Barry sent you the following:
If t = tan(A/2) = ra/s then
a = 2.R.sinA = 4.R.t/(1 + t^2) = 4.R.s.ra/(s^2 + ra^2) (1)
bc = (abc)/a = 2.R.S/a = 4.R.(s-a)ra/a
Hence
ab + ac + bc = a(b+c) + bc = a(2s-a) + 4.R.(s-a)ra/a
or substituting a from (1) we get Francisco's formula.
Hence Barry sent you the proof of Francisco's formula.
Best regards