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Math Help - Geometry

  1. #1
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    Exclamation Geometry

    Prove that the 9-point circle bisects every segment connecting the orthocenter to a point on the circumcircle.

    Thanks very much for your help!!!!
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  2. #2
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    Quote Originally Posted by suedenation
    Prove that the 9-point circle bisects every segment connecting the orthocenter to a point on the circumcircle.

    Thanks very much for your help!!!!
    Another way of saying this is that the radius of the nine-point circle is half the radius of the circle circumscribing the triangle. Now, the 9 point circle can be thought of the circumscribing triangle of the midpoint of this triangle. Because if you have 3 non-colieanr points you can only draw a unique circle around them.
    -----
    Draw triangle ABC, then, mark off its midpoints XYZ. Draw triangle XYZ, we can show that XYZ is similar to ABC, by ratio of 2:1. Next, by the Extended law of sine, we know that 2R=\frac{a}{\sin \alpha} for the big triangle and  2r=\frac{x}{\sin X} for the small triangle. But the sides are in proportion of 2:1 thus, 2x=a but thet are similar thus, \sin \alpha= \sin X. From here we see 2r=R.

    Any problems? I explained it horrifically.
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