Circumcircles
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 Circumcircles The circumcircle or circumscribed circle of a triangle(or a polygon) is a circle which passes through all the vertices of the triangle. The center of the circumcircle is called the circumcenter, and the circle's radius is called the circumradius.
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Example: ABCD is a parallelogram whose diagonals meet in O. Prove that the circumcircles of AOB and COD touch each other.
 Solution: Draw circumcircle of AOB and draw tangent SOT to this circle at O. In order to prove that the circumcircles of AOB and COD touch each other at O, it is sufficient to show that SOT is a tangent to the circumcircle of  COD. Draw circumcircle of  COD. Now, SOT is tangent to the circumcircle of AOB and OA is a chord through O.   SOQ = OBA     [Angles in alternate segments are equal ] but,  OBA = TOC     [Vertically opposite angles] since AB | | DC and BD cuts them.         DBA = BDC     [Alternate angles] or,    OBA = ODC from (i) and (ii) ,you get          ODC = TOC But, these are angles in alternate segments for the circle circumscribing OCD. So, SOT is also a tangent to this circle. Thus, SOT is a common tangent to the two circles. Hence, circumcircles of triangles AOB and COD touch each other at O.

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