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GeoGebra

Triangle Centers

Autor:talljerome
The "center" of a triangle may be defined in many ways. In fact, an Encyclopedia of Triangle Centers catalogs tens of thousands of ways of defining a triangle's "center." This construction presents the four centers that are most commonly encountered in a Geometry class:
  • Incenter - Intersection point of angle bisectors
  • Circumcenter - Intersection point of perpendicular bisectors
  • Centroid - Intersection point of medians
  • Orthocenter - Intersection point of altitudes
Each of these four centers can be found with simple straightedge-and-compass constructions. While there is still great value in the "old school" paper and pencil approach, dynamic geometry software allows us to observe how things change (or not) while the triangle is modified before our eyes. For ΔABC below, select one of the four centers from the pull-down menu. Drag the vertices A, B, and C and observe.
Incenter To help remember the name of this center, envision the circle inside the center of this triangle. At the start of the animation, angle bisectors evenly divide each vertex angle of ΔABC. Six line segments then slide out from vertices A, B, and C. At each vertex of ΔABC, we then see a pair of smaller triangles has been created.

Given that each of the sliding segments is perpendicular to the triangle side it intersects, which triangle comparison can be used to prove that the pair of right triangles meeting at any given vertex are congruent?

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  • A
  • B
  • C
  • D
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By the end of the animation, the sliding segments have become radii of the inscribed circle. Those radii and the angle bisectors form six acute angles at the center of the circle. How do the measures of those central angles compare to the measures of the vertex angles of ΔABC?

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Circumcenter To help remember the name of this center, recall that the circumference of a circle measures the distance around its edge, and that circular edge passes through each vertex A, B, C of the triangle. During the animation, six line segments emerge from the sides of ΔABC. Bordering each side, a pair of smaller triangles is created.

Drawn from the midpoint of each side of ΔABC, the dashed line segment lies along the perpendicular bisector of that side. Which triangle comparison can be used to prove that the pair of right triangles along any given side are congruent?

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  • A
  • B
  • C
  • D
Verifique minha resposta (3)

The circumcenter is the center of the circumscribed circle. Drag one or more vertices A, B, C in the plane until the circumcenter lies on one side of ΔABC. I. What kind of triangle does ΔABC appear to be when this condition occurs? Drag one or more vertices A, B, C in the plane until the circumcenter lies outside of ΔABC. II. What kind of triangle does ΔABC appear to be when this condition occurs?

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Centroid This vocabulary term may not offer great clues to its meaning. But maybe it makes you think of an "android," or humanoid robot... something whose construction relies heavily on the understanding of physics. This may bring to mind that the centroid is central to an important physical property of any triangle.

At the end of the animation, an image of a pencil swoops over to the centroid. What physical property do you think this is intended to bring to mind? Consider cutting a triangle out of a piece of cardboard (or stiff cardstock paper) and testing this out for yourself.

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The animation constructs the three medians of ΔABC, which connect each vertex to the midpoint of its opposite side. The centroid divides each median into two segments. The length of the longer segment turns out to maintain a constant ratio with the length of the smaller segment. What does that ratio appear to be? Feel free to hold a ruler up to the screen.

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Orthocenter This vocabulary term offers a clues to its meaning if you know that the prefix "ortho" in a geometric context suggests that some objects are oriented at right angles to each other. Accordingly, an altitude is a line segment that joins a triangle vertex to its opposite side, forming a right angle with that side. (It's worth noting that the construction of a circumcenter also involves segments intersecting at right angles, so this should not be confused with the orthocenter.)

Drag one or more vertices A, B, C in the plane until the orthocenter lies on one vertex of ΔABC. I. What kind of triangle does ΔABC appear to be when this condition occurs? Drag one or more vertices A, B, C in the plane until the orthocenter lies outside of ΔABC. II. What kind of triangle does ΔABC appear to be when this condition occurs?

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Toggle on the "show all 4 centers" switch and drag one or more vertices of ΔABC around the plane. Observe a geometric relationship that will always hold true for three of the four centers. What is that relationship? Which three of the four centers are included in this relationship? Do a web search as needed and find: Who is the famous Mathematician who is credited with discovering this relationship?

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