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Prove and apply properties of angle bisectors and perpendicular bisectors of a triangle.

Objectives. Prove and apply properties of angle bisectors and perpendicular bisectors of a triangle. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

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Prove and apply properties of angle bisectors and perpendicular bisectors of a triangle.

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  1. Objectives Prove and apply properties of angle bisectors and perpendicular bisectors of a triangle.

  2. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

  3. _________ - A circle in a polygon that intersects each line that contains a side of the polygon at exactly one point. ______________ - A circle that contains all the vertices of a polygon The incenter is the center of the triangle’s inscribed circle. The circumcenter of ΔABC is the center of its circumscribed circle.

  4. MZ is a perpendicular bisector of ∆GHJ. Check It Out! Example 1a Use the diagram. Find GM.

  5. Check It Out! Example 1b Use the diagram. Find JZ.

  6. Check It Out! Example 2 Find the circumcenter of ∆GOH with vertices G(0, –9), O(0, 0), and H(8, 0) .

  7. Remember! The distance between a point and a line is the length of the perpendicular segment from the point to the line.

  8. Unlike the circumcenter, the incenter is always inside the triangle.

  9. QX and RX are angle bisectors of ΔPQR. Find the distance from X to PQ. Check It Out! Example 3a

  10. Check It Out! Example 3b QX and RX are angle bisectors of ∆PQR. Find mPQX.

  11. Example 4: Community Application A city planner wants to build a new library between a school, a post office, and a hospital. Draw a sketch to show where the library should be placed so it is the same distance from all three buildings.

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