Download Circumscribed Circles Definition. The circumscribed circle or of a

Circumscribed Circles
Definition. The circumscribed circle or of a polygon
is a circle which passes through all the vertices of the
polygon. The center of this circle is called the circumcenter.
Definition. A polygon which can be inscribed in a circle is called a cyclic polygon.
Fact: All regular simple polygons, all triangles and all
rectangles are cyclic.
How to find the circle that circumscribes a triangle:
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1.3.3 Quadrilaterals
Every Simple quadrilateral has at least one diagonal
through the interior of the quadrilateral.
Theorem 1.3.10: The sum of the interior angles of a
quadrilateral is 360◦ .
Proof:
Definition. A quadrilateral that is inscribed in a circle
is called a cyclic quadrilateral (or sometimes a concyclic quadrilateral).
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Theorem 1.3.11: Let ABCD be a simple cyclic quadrilateral. Then
1. Each opposite interior angle pair sums to 180◦ .
2. Each exterior angle is congruent to the opposite
interior angle.
Proof: Homework question.
Theorem 1.3.12: Let ABCD be a simple quadrilateral.
If the opposite angles sum ot 180◦ , then ABCD is a
cyclic quadrilateral.
Proof: Homework question.
Example 1.3.13, Simpson’s Theorem: Given 4ABC
and a point P on its circumcircle, the perpendiculars
dropped from P meet the sides of the triangle in three
collinear points.
The line is called the Simpson line corresponding to P .
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