Circle inscribed in a semicircle

spoon737 · #1 $Old$ June 29th, 2009, 12:04 PM

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?

earboth · #2 $Old$ June 29th, 2009, 12:46 PM

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.