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Math Help - triangles, midpoints and parallelism

  1. #1
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    triangles, midpoints and parallelism

    Let ABC be an triangle. On AB side of the triangle let D, E be two points so that AD\leq AE. Let DF \parallel BC,F\in AC and EG\parallel AC,G\in BC. Let L , M , N , P , R be the midpoints of the segments: [CF] , [FD] , [DE] , [EG] , [GC]. Proof that [CN] , [PL] , [MR] can be the three sides of a triangle. Find a condition so that MP\parallel AB.
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    Re: triangles, midpoints and parallelism

    triangles, midpoints and parallelism-tri.png
    With out any loss of generality we consider an oblique co-ordinate system with the origin as A with B and C along x and y axis such that BC=a, AC=b, AB=c the lengths of the given triangle. Now say AD = \alpha AB, BE = \beta AB where \alpha , \beta are some constants. Using the condition AD \le AE we have \alpha + \beta \le 1.

    Using Basic Proportionality Theorem in this triangle and the formula for internal division of a line segment according to a given ratio. We can determine the co-ordinates of the points as F(0,\alpha b), L(0,\frac{\alpha+1}{2}b), R(\frac{1- \beta}{2}c, \frac{1+\beta}{2}b), G((1-\beta )c, \beta b), D(\alpha c, 0),

    E((1-\beta)c,0), M(\frac{\alpha}{2} c, \frac{\alpha}{2} b), N(\frac{1- \beta + \alpha}{2}c,0), P((1-\beta)c, \frac{\beta}{2}b).

    Use the Cosine Rule to get the inclination between the co-ordinate axes say \theta as a^2 = c^2 + b^2 - 2bc \cos \theta.Now use the distance formula in the oblique system to determine the lengths MR, CN, PL and show that the sum of lengths of any two sides is greater than the third side so they can form a triangle. \alpha + \beta \le 1 will prove to be helpful.

    Use the slope equation to get the condition between \alpha , \beta for MP \parallel AB.

    Kalyan
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  3. #3
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    Re: triangles, midpoints and parallelism

    This partial solution is similar to the above, but the algebra for the first question was beyond me. As implied above, MP is parallel to AB iff length AD = length EB. I put the given figure in a dynamic geometry program which was very helpful. Here's the image.
    triangles, midpoints and parallelism-mhftriangle1.png
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  4. #4
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    Re: triangles, midpoints and parallelism

    As implied above, MP is parallel to AB iff length AD = length EB.




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