Question 1124087
.


            This problem is above the average level of school Math problems.


            It is  the level of a Math circle.


            It requires combining several ideas and formulas.
 


<pre>
1.  Calculate the area of the triangle via  inradius "r" and  semi-perimeter "s" in this way:

        Area = r*s.                                          (1)

    It gives you  Area = 1*7 = 7 square units.



2.  Use the Heron's formula for the area:

        Area = {{{sqrt(s*(s-a)*(s-b)*(s-c))}}},    which gives you

        7 = {{{sqrt(7*(7-a)*(7-b)*(7-c))}}}.


    Square both sides to get

       7^2 = 7*(7-a)*(7-b)*(7-c).


    Cancel the factor 7 in both sides

       7 = (7-a)*(7-b)*(7-c).


       7 = (49 - 7a - 7b + ab)*(7-c) = 

         = 343 - 49a - 49b + 7ab - 49c + 7ac + 7bc - abc = 

         = 343 - 49*(a + b + c) + 7*(ab + bc + ac) - abc.     (2)


3.  You are given the semi-perimeter  s = 7,  so you know the perimeter too:

         a + b + c = 7*2 = 14.                                (3)


    Substitute it into the formula (2) to get

         7 = 343 - 49*14 + 7*(ab + bc + ac) - abc.            (4)


4.  To find abc, use the formula for the area of a triangle 

       Area = {{{(abc)/(4*R)}}},   where R is the circumradius             (5)


    Substituting the given and known data, it gives you

       7 = {{{(abc)/(4*3)}}},    or    abc = 7*4*3 = 84.                   (6)


5.  Substitute the found value of abc into (4) to get       

       7 = 343 - 49*14 + 7*(ab + bc + ac) - 84.


    Simplify

       ab + bc + ac = {{{(7- 343 + 49*14 + 84)/7}}} = 62.                   (7)


6.  Now you are in one step from getting the answer.

    You have 

        a + b + c = 14.


    Square it:

        (a + b + c)^2 = 14^2 = 196 = a^2 + b^2 + c^2 + 2*(ab + ac + bc),

    or

        a^2 + b^2 + c^2 = 196 - 2*(ab + ac + bc) = 196 - 2*62 = 72.


<U>Answer</U>.  a^2 + b^2 + c^2 = 72.
</pre>


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On formula &nbsp;(1)&nbsp; see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-area-of-a-triangle-via-the-radius-of-the-inscribed-circle.lesson>Proof of the formula for the area of a triangle via the radius of the inscribed circle</A> 

in this site.


On Heron's formula see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/-Proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>Proof of the Heron's formula for the area of a triangle</A>, 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/One-more-proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>One more proof of the Heron's formula for the area of a triangle</A>, 

in this site.


On formula &nbsp;(5)&nbsp; see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A> 

in this site. 


Also, &nbsp;you have this free of charge online textbook on Geometry

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A> 

in this site.


The referred lessons are the part of this online textbook under the topic "<U>Area of triangles</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in GEOMETRY
https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson


to your archive and use it when it is needed.