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.

Circle I is said to be inscribed in triangle XYZ.

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.

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