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 Build a triangle. (Posted on 2012-03-15)
The two sides of a triangle (a and b) are given: b≤a≤2b. The medians to those sides are orthogonal. Build this triangle.

 See The Solution Submitted by Ady TZIDON No Rating

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`ANALYSIS:`
`Let ABC be the desired triangle with standardside lengths of a, b, and c. Let AA' abd BB'be the medians intersecting orthogonally atpoint I. Let |AA'| = 3m and |BB'| = 3n. Aroundpoint I we have three right triangles AIB, A'IB, and AIB'. Using the Pathagorean theorem:`
`   AIB: c^2 = 4(m^2 + n^2)               (1)  A'IB: a^2 = 4(m^2 + 4n^2)              (2)  AIB': b^2 = 4(4m^2 + n^2)              (3)`
`Combining (2) and (3) gives`
`  a^2 + b^2 = 20(m^2 + n^2)              (4)`
`Combining (1) and (4) gives`
`  a^2 + b^2 = 5c^2                       (5)`
`CONSTRUCTION:`
`Using lengths a and b construct a length csatifying (5). Let C(P,r) denote a circlewith center P and radius r. WOLOG we willassume that a <= b.`
`Construct a diameter CD of C(B,a). Constructpoints E and F on the same side of CD suchthat both line segments CE and DF areperpendicular to CD with |CE| = b and |DF| = 2a. Construct line segments BE, BF, andEF. Line segment BF intersects C(B,a) atpoint G. Construct point on line segment BEsuch that line segments GH and EF are parallel.Let A be the point of intersection of C(B,|BH|)and C(C,b) on the opposite side of CD from point G. The desired triangle is ABC.`
`Note: While checking out the above construction      with Geometer's SketchPad it was found      that b must be less than 2a. This is a      consequence of equation (5) and the      triangle inequality b < a + c.`
`PROOF:`
`We have to prove that |AB| = c satisfies (5).`
`  |AB| = |BH|          by construction`
`   |BH|     |BE|  ------ = ------      triangles BHG and BEF    |BG|     |BF|       are similar by con-                       struction.`
`Therefore,`
`                   |BE|^2  |BH|^2 = |BG|^2 --------                   |BF|^2`
`                a^2 + b^2         = a^2 -----------                  5a^2`
`         or`
`  a^2 + b^2 = 5|BH|^2 = 5|AB|^2`
`QED `

 Posted by Bractals on 2012-03-15 12:45:48

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