Unit 7, Lesson 11
Circles in Triangles
Integrated Math 2
11.1
Warm-up
Use a compass to draw the largest circle possible that fits inside each triangle.
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Use a compass to draw the largest circle possible that fits inside each triangle.
The image shows an equilateral triangle . The angle bisectors are drawn. The incenter is plotted and labeled .
Prove that the incenter is also the circumcenter.
We have seen that the incenter of a triangle is the same distance from all three sides of the triangle. If we draw the congruent segments representing the shortest distances from the incenter to the triangle’s sides, we can think of them as radii of a circle centered at the incenter. This circle is the triangle’s inscribed circle.
In this diagram, segments and are angle bisectors. Point is the triangle’s incenter, and the circle is inscribed in the triangle.
The inscribed circle is the largest possible circle that can be drawn inside a triangle. Also, the three radii that represent the distances from the incenter to the sides of the triangle are by definition perpendicular to the sides of the triangle. This means the circle is tangent to all three sides of the triangle.
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Triangle ABC with a circle drawn in the center. Center of the circle is point D. Line segment DG, DF, and DE are equivalent. Angles BGD, BFD and CED = 90 degrees. Dashed lines from each point A, B, C to the center D.