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Thread: Circle inscribed in a semicircle

  1. #1
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    Circle inscribed in a semicircle

    Take a look at this picture:

    Proposed Geometry Problem 310: Circle Inscribed in a Semicircle, 45 Degrees Angle. College Geometry, SAT Prep. Elearning

    I'm not trying to solve the problem in the link, but the picture is relevant to what I'm trying to figure out. If I have a circle inscribed in a semicircle like that, is it true that the points O, C, and E are collinear? If so, how can I prove it?
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  2. #2
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    Quote Originally Posted by spoon737 View Post
    Take a look at this picture:

    Proposed Geometry Problem 310: Circle Inscribed in a Semicircle, 45 Degrees Angle. College Geometry, SAT Prep. Elearning

    I'm not trying to solve the problem in the link, but the picture is relevant to what I'm trying to figure out. If I have a circle inscribed in a semicircle like that, is it true that the points O, C, and E are collinear? If so, how can I prove it?
    1. The semi-circle and the inscribed circle have a common tangent in E. The radius of a circle is perpendicular to the tangent in the tangent point. Since OE is a radius and CE is a radius OE and CE must be the same line. Thus the three points are collinear.

    2. To find the radius of the inscribed circle draw the tangent at the semi-circle in E. The tangent crosses the line AB in F. Draw the angle bisector of $\displaystyle \angle(EFO)$. The angle bisector cuts OE in C. CE is the radius in question.

    3. There is at least one property missing to prove that $\displaystyle \alpha = 45^\circ$. This construction actually asks you to construct a symmetric kite.
    Attached Thumbnails Attached Thumbnails Circle inscribed in a semicircle-tangcircle_insemicircle.png  
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