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Re: TRIANGLE CONSTRUCTION A, a, Sum_of _altitudes

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  • Antreas Hatzipolakis
    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]
    • Luís Lopes
      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.

        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]


















        [Non-text portions of this message have been removed]
      • Antreas
        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, 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
        • Luís Lopes
          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,

            > 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



















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