Preliminary: First you must understand the relationship between inscribed and circumscribed, for example:

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1 10.7 Inscribed and Circumscribed Polygons Lesson Objective: After studying this section, you will be able to: Recognize inscribed and circumscribed polygons Apply the relationship between opposite angles of an inscribed quadrilateral Identify the characteristics of an inscribed parallelogram Preliminary: First you must understand the relationship between inscribed and circumscribed, for example: Example 1: A given triangle can be an inscribed polygon because a circle can be circumscribed through its vertices. Example 2: A circle can be inscribed in a given triangle since a circle can be drawn tangent to its sides. The given triangle would be circumscribed about the circle. Definitions: Inscribed in When a polygon is inscribed in a circle, all of its vertices lie ON the circle. (see example 1 above) Circumcenter The center of a circle circumscribed about a polygon is the circumcenter of the polygon. (see example 1 above) Circumscribed about A polygon is circumscribed about a circle if each of its sides is tangent to the circle. (see example 2 above) Incenter The center of a circle inscribed in a polygon is the incenter of the polygon. (see example 2 above)

2 Therefore: If a polygon is inscribed in a circle, then all of its vertices lie ON the circle, and circumcenter is the name used for the center of the circle that is circumscribed about the polygon If a polygon is circumscribed about a circle, then all of its sides are tangent to the circle, and incenter is the name used for the center of the circle that is inscribed in the polygon. Be able to construct each of the following: Construct a circle circumscribed about a given triangle How to construct a circle about a triangle: 1. First construct the perpendicular bisectors of two sides of the triangle, 2. Then use the intersection of these as the center (circumcenter) 3. Finally, construct a circle whose radius equals the distance to one of the three vertices. Construct a circle inscribed in a given triangle How to construct a circle inscribed in a given triangle: 1. Construct the bisectors of two of the angles 2. Drop a perpendicular from the intersection of the two angle bisectors 3. Use the perpendicular distance as the radius of the circle and the point of intersection as your center (incenter). KEY: bisectors: BLUE segments circumcenter of : BLUE radii of circumscribed, RED dashed segments Do you see the circle taking shape around the triangle? KEY: bisectors: BLUE rays Perpendicular drawn from intersection of angle bisectors: GOLD segment at base side of Δ. Radii of inscribed equals the perpendicular distance from incenter of polygon: RED dashed segments are equal to length of gold segment. Do you see the circle taking shape inside the triangle?

3 Construct tangents from an exterior point to a circle Given: O with point P as an exterior point A O P B Step 1: Draw segment OP Step 2: Construct the perpendicular bisector of OP Step 3: Using the midpoint of OP as the center and a radius equal to half of OP, construct a second circle. (The new circle should intersect both points P and O) Step 4: Place points at the intersections of your two circles (labeled A and B) Step 5: Draw segments PA and PB Step 6: Draw radii OA and OB Notice that PA and PB are tangents to O because any line drawn to a circle such that it intersects a radius is perpendicular to the radius. PA and PB are lines drawn to radii OA and OB from a point (point P) outside the circle containing both radii. The intersection of line PA with OA and PB with OB are called the points of tangency!!eureka! Another way to prove that the tangent lines intersect the radii of the original circle at 90 degree angles is to notice that PAO and PBO are both inscribed angles of the 2nd circle on SEMICIRCLES! Questions to ponder: Is it possible to inscribe a circle in a parallelogram? (why/why not?) Is it possible to circumscribe a parallelogram about a circle? (why/why not?)

4 BEWARE! Not all polygons are inscribable or circumscribable. A test for circumscribing a circle about a quadrilateral is given in Theorem 93, which you will test later in the notes. Argument! In a polygon, any three vertices determine the circumscribed circle. If the remaining vertices lie on that circle, then the polygon can be inscribed in a circle. Triangles ARE polygons, but they are NOT quadrilaterals! Theorem 128: (Ch 14) The perpendicular bisectors of the sides of a triangle all intersect at one point that is equidistant from the vertices of the triangle. (The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle. The distance from the circumcenter to any one of the three vertices of the triangle establishes the radius of the circle that can be circumscribed about the triangle! Key: A Lines l, m, and n bisect the sides of the given triangle. n m BLUE point in center is the circumcenter. B l C BLUE segments from circumcenter have equidistance to the vertices of the given triangle and they establish radii of circle. Circumscribed circle! Isn t it easy to see that every triangle can be inscribed in a circle since ALL triangles have three vertices whose sum is always equal to the measure of a semicircle! (180⁰) Theorem 129: (Ch 14) The bisectors of the angles of a triangle always intersect at a single point that is equidistant from the sides of the triangle (The point of concurrency of the angle bisectors is called the incenter of the triangle). The incenter of the triangle determines the center of a circle that can be inscribed in the triangle.

5 QUESTION! Is this the case for ALL polygons? Answer: NO! Remember from above: In a polygon, any three vertices determine the circumscribed circle. If the remaining vertices lie on that circle, then the polygon can be inscribed in a circle. Theorem 93: If a quadrilateral is inscribed in a circle, its opposite angles are supplementary. TEST to see whether each of the quadrilaterals below can be inscribed in a circle: Instructions - Place a fourth dot where it belongs to complete each quadrilateral below. Quadrilateral Parallelogram Rhombus Rectangle Square Kite Trapezoid Isosceles Trapezoid Question: What quadrilaterals were you able to complete? Not able to complete? Question: What is the sum of the angles of every quadrilateral? Question: Do all quadrilaterals possess opposite angles that are supplementary? Question: Of the quadrilaterals you were able to complete, what do they all have in common? Theorem 94: If a parallelogram is inscribed in a circle, it MUST be a rectangle. *** Note - Remember squares are types of rectangles! One last thing, this time you are to only to inscribe kites in each of the circles below: Question: What seems to be true about each of the kites you were able to inscribe above? (answer at end of lesson)

6 Communicating Mathematics: Draw a diagram to illustrate each of the following statements: 1. If a figure is inscribed, then the circle is circumscribed. 2. If the figure is circumscribed, then the circle is inscribed. Additionally, the vertices of an inscribed polygon are called concyclic. This term is used in problems 22 and 23 on page 491. Answer to inscribed kites question: You should have observed that the kites you were able to inscribe always had at least one pair of right angles opposite each other!