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    prove that the circumcenter of a given triangle is the orthocenter of the triangle formed by joining the midpoints of the sides of the original triangle. thanks for ur help
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  2. #2
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    Quote Originally Posted by peri123 View Post
    prove that the circumcenter of a given triangle is the orthocenter of the triangle formed by joining the midpoints of the sides of the original triangle. thanks for ur help
    The original triangle is ABC.

    The mid-parallels to the sides of the triangles form the new triangle.

    The mid-parallel m_a is parallel to a (drawn in red). The perpendicular bisector of a is perpendicular to m_a too. The midpoint of a is the vertex of the new triangle.

    The perpendicular bisectors of ABC cut themselves in P. Since the perp. bisectors pass through the vertices of the new triangle and are perpendicular to the sides of the new triangle they are simultaneously heights in the new triangle. Since the bisectors of the original triangle intersect at P the heights of the new triangle intersect in P too.
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    Last edited by earboth; August 4th 2008 at 11:09 PM.
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