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Math Help - 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 \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 \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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