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Inscribed & Circumscribed Polygons

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Inscribed & Circumscribed Polygons

Lesson 10.7

A polygon is inscribed in a circle if all of its vertices lie on the circle.

A polygon is circumscribes about a circle if each of its sides is tangent to the circle.

- The center of a circle circumscribed about a polygon is the circumcenter of the polygon.

O

C

P is the circumcenter of rectangle COME

P

E

M

- The center of a circle inscribed in a polygon is the incenter of the polygon

P

S

O is the incenter of hexagon SPRING!

O

G

R

I

N

- If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

Once there was a plain old parallelogram named Rex Tangle. Rex was always trying to fit in – into a circle, that is. One day when he awoke, he found that he had straightened out and was finally able to inscribe himself. What had the plain old parallelogram turned into?

Theorem 94: If a parallelogram is inscribed in a circle, it must be a rectangle.

If ABCD is an inscribed parallelogram, then…

- BD and AC are diameters.
- O is the center of the circle.
- OA, OB, OC and OD are radii.
- (AB)2 + (BC)2 = (AC)2, and so forth.

ABCD is inscribed in circle O.

B supp. ADC

ADC supp. ADE

B ADE

Given

If a quadrilateral is inscribed in a circle, its opposite s are supp.

Two s forming a straight are supp.

Two s supp. to the same are .