## Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

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• 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
Message 1 of 5 , Feb 1, 2011
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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]
• 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
Message 2 of 5 , Feb 1, 2011
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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.

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]

[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 +
Message 3 of 5 , Feb 1, 2011
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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
Message 4 of 5 , Feb 16, 2011
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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,

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]
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