**EMAT
6700**

**by
Brad Simmons**

**Inscribed and Circumscribed Circles**

Use geometer's sketchpad construct inscribed and circumscribed circles.

**Inscribed Circles**

The three angle bisectors of a triangle
intersect in a single point called the **incenter**. This point is the
center of a circle that touches the three sides of the triangle. Except
for the three points where the circle touches the sides, the circle is inside
the triangle. This circle is said to be **inscribed** in the triangle.

1. Begin with triangle XYZ.

2. Construct the bisectors of angle X and angle Z. Label the point the bisectors intersect point I.

3. Construct a perpendicular segment from point I to segment XZ. Label the point of intersection with segment XZ point A.

4. Select point I and segment AI. Construct circle by center and radius.

Please click here for a geometer's sketchpad sketch of the figure shown above.

**Construct the inscribed circle for triangle PQR shown below.**

**Circumscribed
Circles**

The three
perpendicular bisectors of the sides of a triangle also meet in a single
point. The point called the **circumcenter** is the center of the
circumscribed circle called the **circumcircle**, which passes through each
vertex of the triangle. Except for the three points where the circle
touches the triangle, the circle is outside the triangle.

1. Begin with triangle XYZ.

2. Construct the perpendicular bisectors of segment XY and segment XZ. Label the point of intersection of the bisectors point C.

3. Construct segment CX. Select point C and segment CX. Construct circle by center and radius.

Circle C is the circumcircle of triangle XYZ. In other words, circle C is the circumsrcibed circle for triangle XYZ.

Please click here for a geometer's sketchpad sketch of the figure shown above.

**Construct the circumscribed circle for
triangle FGH shown below.**