Concurrent Lines, Medians, and Altitiudes

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Concurrent Lines, Medians, and Altitiudes
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Concurrent Lines

• When three or more lines intersect in one point,
  they are concurrent.
• Point of Concurrency the point at which the
  lines intersect.
• For a triangle, there are four different sets of
  concurrent lines.

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Circumcenter

• The perpendicular bisectors of the sides of a
  triangle are concurrent (all intersect) at a
  point equidistant from the vertices.
• The point of concurrency is called the
  circumcenter of the triangle.

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Picture of the Circumcenter
CH CJ CG
C is the circumcenter.
Using a compass with the point at C, a circle can
be drawn that passes thru G, H and J. The circle
is circumscribed about the triangle.
C
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Incenter

• The bisectors of the angles of a triangle are
  concurrent (all intersect) at a point equidistant
  from the sides.
• The point of concurrency is called the incenter
  of the triangle.

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Picture of the Incenter
IY IR IW
I is the incenter.
Using a compass with the point at I, a circle can
be drawn that passes thru Y, R and W. The circle
is inscribed in the triangle.
I
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Medians

• Median of a Triangle A median is a segment
  drawn from the vertex of a triangle to the
  midpoint of the opposite side.

Median
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Centroid

• The medians of a triangle are concurrent (all
  intersect) at a point that divides each median
  into two segments, one of which is twice as long
  as the other.
• The point of concurrency is called the centroid.
• Also called the center of gravity or center of
  balance.

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Picture of Centroid
2x
8
k
5
b
C
2b
2k
10
x
4
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Altitudes

• Altitude of a Triangle An altitude is a segment
  drawn from the vertex of a triangle perpendicular
  to the opposite side.
• Can be inside, on a side, or outside the traingle

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Orthocenter

• The lines that contain the altitudes of a
  triangle are concurrent (they all intersect at a
  single point).
• The point of concurrency of the altitudes is
  called the orthocenter.

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Orthocenter
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Finding Lengths of Medians

• M is the centroid of DWOR, and WM 16. Find WX.

WX 24
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Identifying Medians and Altitudes

• Is KX a median, an altitude, neither, or both?

Both