You need a makeshift compass made from string and pencil. Use these materials to construct the perpendicular bisectors of the three sides of the triangle below. How did using this tool differ from using a compass and straightedge?

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Compare your construction with that of your partner. Did you obtain the same results?

When three or more lines intersect in a single point, they are _____________________.

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When three or more lines intersect in a single point, they are concurrent, and the point of intersection is the _____________________________.

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The point of concurrency of the three perpendicular bisectors is the _________________________________________.

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The circumcenter of is shown below as point P.

P is equidistant from A and B since it lies on the ________________ of .

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The circumcenter of is show below as point P.

P is also ________________ from B and C since it lies on the perpendicular bisector of .

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The circumcenter of is shown below as point P.

Hence, π΄π = π΅π = πΆπ, which suggests that π is the point of _______________________ of all three perpendicular bisectors.

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The circumcenter of is shown below as point P.

You have also worked with angle bisectors. The construction of the three angle bisectors of a triangle also results in a point of concurrency, which we call the _______________________.

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Use the triangle below to construct the angle bisectors of each angle in the triangle to locate the triangle's incenter.

Draw your drawing on paper, take a picture, and upload it using the image upload icon .

If you do not have the ability to upload an image of your work, type "Drawing is on paper."

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Earlier in this lesson, we explained why the perpendicular bisectors of the sides of a triangle are always concurrent. Using similar reasoning, explain clearly why the angle bisectors are always concurrent at the incenter of a triangle.

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Point π΄ is the __________________________ of . (Notice that it can fall outside of the triangle).

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On a separate piece of paper, draw two triangles of your own below and demonstrate how the circumcenter and incenter have these special relationships.

Draw your drawing on paper, take a picture, and upload it using the image upload icon .

If you do not have the ability to upload an image of your work, type "Drawing is on paper."

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How can you use what you have learned in Exercise 3 to find the center of a circle if the center is not shown?

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