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Thread: Area of a Triangle Formula: inradius & semiperimeter

  1. #1
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    Area of a Triangle Formula: inradius & semiperimeter

    How would you show a triangle's area is equal to the product of its inradius and its semiperimeter?

  2. #2
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    What is an inradius?.

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    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

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    I learned something there. I had never heard of an inradius.

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    Re: Area of a Triangle Formula: inradius & semiperimeter

    Quote Originally Posted by Trenters4325
    How would you show a triangle's area is equal to the product of its inradius and its semiperimeter?

    The internal circle tangent to the three sides and the incenter as center.
    The radius of the inscribed circle is r = A/s where A = the area of the triangle and s
    = the semi-perimeter = (a + b + c)/2, a, b, and c being the three sides.
    The radius of the inscribed circle may also be derived from r = ab/(a + b + c).
    The radius of the inscribed circle may also be derived from the particular m and n
    used in deriving a Pythagoraen Triple triangle by r = n(m - n).
    If x, y, and z are the points of contact of the incircle with the sides of the triangle
    A, B, C, then Cx = Cy = s - c, Bx = Bz = s - b, and Ay = Az = s - a.
    TchrWill

    No matter how insignificant it might appear, learn something new every day.

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    Re: Area of a Triangle Formula: inradius & semiperimeter

    Quote Originally Posted by TchrWill
    The radius of the inscribed circle is r = A/s where A = the area of the triangle and s...
    Where did A/s come from?

    Also, on the Wolfram website, where did they right side of the first line come from?

  7. #7
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    Well, O.K.
    You have asked a rather advanced question.
    You have been given two rather clear answers.
    Now, we have no way of knowing your background: what theorems you know; how well you understand triangles; how far along in the Euclidean Geometry you are.

    You have got to do somethings for yourself. The questions you have asked seem to indicate a real lack of understanding of the particulars of this problem on your part.
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

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    I'm actually in calculus, so I studied geometry a while ago, and although I understand it pretty well, I'm kind of rusty.

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    Re: Area of a Triangle Formula: inradius & semiperimeter

    Quote Originally Posted by Trenters4325
    How would you show a triangle's area is equal to the product of its inradius and its semiperimeter?
    1--Consider an acute triangle ABC, A lower left, B lower right and C at the top.
    2--Bisect each angle
    3--The intersection of the bisectors is the center of the inscribed circle of the triangle with radius r..
    4--Let the center of this incircle be called O.and the three sides a, b and c.
    5--Consider the three triangles AOB, BOC and COA
    6--The areas of these triangles are cr/2, ar/2 and br/2
    7--Therefore the total area of the triangle ABC is A = cr/2 + ar/2 + br/2
    8--This simplifies to A = r(a + b + c)/2
    9--(a + b + c) is the semi-perimeter of the triangle, s.
    10--Therefore, A = rs.
    TchrWill

    No matter how insignificant it might appear, learn something new every day.

  10. #10
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    Re: Area of a Triangle Formula: inradius & semiperimeter

    Quote Originally Posted by TchrWill
    3--The intersection of the bisectors is the center of the inscribed circle of the triangle with radius r..
    How do you know that the point of intersection is equidistant from a, b, and c?

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