- Solutions:

http://anthrakitis.blogspot.com/2011/02/triangle-construction-a-ha-hb-hc.html

On Fri, Jan 28, 2011 at 12:07 PM, Antreas Hatzipolakis <anopolis72@...> wrote:

[Non-text portions of this message have been removed]

> To construct triangle ABC if are given A, a, h_a + h_b + h_c = h.

>

> APH

>

- Dear Hyacinthists, Antreas,

Very nice, Antreas.

There is a typo in your link. Let h_a + h_b + h_c = h.

Then

bc+ca+ab=2R.h=k^2.

And how about (A,b,h) ?

h_c is known.

Best regards,

Luis

To: Hyacinthos@yahoogroups.com

From: anopolis72@...

Date: Tue, 1 Feb 2011 12:48:19 +0200

Subject: [EMHL] Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

Solutions:

http://anthrakitis.blogspot.com/2011/02/triangle-construction-a-ha-hb-hc.html

On Fri, Jan 28, 2011 at 12:07 PM, Antreas Hatzipolakis <anopolis72@...

> wrote:

[Non-text portions of this message have been removed]

> To construct triangle ABC if are given A, a, h_a + h_b + h_c = h.

>

> APH

>

[Non-text portions of this message have been removed] - Dear Luis

Thanks.

For the problem: A, b, h_a + h_b, I think there is no

R&C construction.

We have this system of equations:

h_a + h_b = bsinC + csinA = bsinC + (bsinCsinA/sinB) =

= (bsinC/sinB)(sinB + sinA)

sinA = sinBCosC + cosBsinC

sin^2B + cos^2B = 1

sin^2C + cos^2C = 1

(four equations with four unknowns: sinB,sinC,cosB,cosC)

Antreas

--- In Hyacinthos@yahoogroups.com, Luï¿½s Lopes <qed_texte@...> wrote:

>

>

> Dear Hyacinthists, Antreas,

>

> Very nice, Antreas.

>

> There is a typo in your link. Let h_a + h_b + h_c = h.

> Then

>

> bc+ca+ab=2R.h=k^2.

>

> And how about (A,b,h) ?

>

> h_c is known.

>

> Best regards,

> Luis

>

>

> To: Hyacinthos@yahoogroups.com

> From: anopolis72@...

> Date: Tue, 1 Feb 2011 12:48:19 +0200

> Subject: [EMHL] Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

>

> Solutions:

>

>

>

> http://anthrakitis.blogspot.com/2011/02/triangle-construction-a-ha-hb-hc.html

>

>

>

> On Fri, Jan 28, 2011 at 12:07 PM, Antreas Hatzipolakis <anopolis72@...

>

> > wrote:

>

>

>

> > To construct triangle ABC if are given A, a, h_a + h_b + h_c = h.

>

> >

>

> > APH - Dear Hyacinthos, Antreas,

Consider three problems:

1) A,a,h_a+h_b+h_c=U

2) A,a,h_a+h_b - h_c=U

3) A,a,h_b+h_c - h_a=U

For 1), following Antreas' first solution,

> http://anthrakitis.blogspot.com/2011/02/triangle-construction-a-ha-hb-hc.html

one has

(b+c)^2 + 4a cos^2(A/2)(b+c) - 8RU cos^2(A/2) - a^2 = 0

One knows (A,a,b+c) and this TC is known. The problem has 1 solution.

For 2), one has

(b-c)^2 + 4a sin^2(A/2)(b-c) + 8RU sin^2(A/2) - a^2 = 0

One knows (A,a,b-c) and this TC is known.

The problem may have 2 solutions: a=5 cos A=11/14 R=7\sqrt(3)/3

b=7 c=8

b=6.885933 c=8.028790

For 3), one has

(b+c)^2 - 4a cos^2(A/2)(b+c) + 8RU cos^2(A/2) - a^2 = 0

One knows (A,a,b+c) and this TC is known.

Is it possible to have data for two solutions?

Best regards,

Luis

To: Hyacinthos@yahoogroups.com

From: anopolis72@...

Date: Tue, 1 Feb 2011 22:40:30 +0000

Subject: [EMHL] Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

Dear Luis

Thanks.

For the problem: A, b, h_a + h_b, I think there is no

R&C construction.

We have this system of equations:

h_a + h_b = bsinC + csinA = bsinC + (bsinCsinA/sinB) =

= (bsinC/sinB)(sinB + sinA)

sinA = sinBCosC + cosBsinC

sin^2B + cos^2B = 1

sin^2C + cos^2C = 1

(four equations with four unknowns: sinB,sinC,cosB,cosC)

Antreas

--- In Hyacinthos@yahoogroups.com, Lu���s Lopes <qed_texte@...> wrote:

>

>

> Dear Hyacinthists, Antreas,

>

> Very nice, Antreas.

>

> There is a typo in your link. Let h_a + h_b + h_c = h.

> Then

>

> bc+ca+ab=2R.h=k^2.

>

> And how about (A,b,h) ?

>

> h_c is known.

>

> Best regards,

> Luis

>

>

> To: Hyacinthos@yahoogroups.com

> From: anopolis72@...

> Date: Tue, 1 Feb 2011 12:48:19 +0200

> Subject: [EMHL] Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

>

> Solutions:

>

>

>

> http://anthrakitis.blogspot.com/2011/02/triangle-construction-a-ha-hb-hc.html

>

>

>

> On Fri, Jan 28, 2011 at 12:07 PM, Antreas Hatzipolakis <anopolis72@...

>

> > wrote:

>

>

>

> > To construct triangle ABC if are given A, a, h_a + h_b + h_c = h.

>

> >

>

> > APH

[Non-text portions of this message have been removed]