Unit 7, Lesson 6
A Special Point
Geometry
6.1
Warm-up
Notice and Wonder: Salt Pile
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
Here is an angle
Two angle bisectors have been constructed in triangle
Triangle ABC with intersecting lines inside. Triangle ABC is intersected by line BD. Angles ABD and CBD are equivalent. Triangle ABC is intersected by line CE. Angles ACE and BCE are equivalent. Point G is the the intersection point of lines BD and CE.
Salt piles up in an interesting way when poured onto a triangle. Why does that happen?
As the salt piles up and reaches a maximum height, new grains of salt will fall off toward whichever side of the triangle is closest. We can show that points on an angle bisector are equidistant from the rays that form the angle. So salt grains that land on an angle bisector will balance and not fall toward either side. This is why we see ridges form in the salt.
As we might conjecture from the salt example, all three angle bisectors in a triangle meet at a single point, called the triangle’s incenter. To see why this is true, consider any two angle bisectors in a triangle. The point where they meet is the same distance from the first and second sides, and is also the same distance from the second and third sides. Therefore, the point is the same distance from all sides, so the third angle bisector must also go through this point.
In the images, segments
Angle QRS bisected by line T. Line from point T to line Q creates 90 degree angle. Line from point T to line S creates 90 degree angle. These segments are equivalent.
The incenter of a triangle is the intersection point of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.
In this diagram the incenter is point
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