Concurrency Theorems:
Concurrent lines are three or more lines that intersect at the same point, known as the
point of concurrency.
In a triangle, the following sets of lines are concurrent:
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The 3 medians the 3 altitudes The perpendicular bisectors of the 3 sides The 3 angle bisectors Concurrency of Angle Bisectors:
The three angle bisectors of a triangle are concurrent at a point equidistant from the
sides of the triangle, known as the incenter of the triangle.
The incenter is the center of a circle inscribed in a triangle.
Concurrency of Altitudes:
The altitudes of a triangle are concurrent in a point called the orthocenter.
Some key facts:
• In an acute triangle, the orthocenter lies in the interior of the triangle.
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In a right triangle, the orthocenter is at the vertex of the right angle.
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In an obtuse triangle, the altitudes and a side need to be extended so
that the orthocenter falls in the exterior of the triangle.
Concurrency of Medians:
The three medians of a triangle are concurrent at a point called the centroid
of the triangle. The distance from each vertex to the centroid is two-thirds
of the length of the entire median drawn from that vertex. In other words, the centroid
divides the medians into a 2 to 1 ratio.
Concurrency of Perpendicular Bisectors:
The three perpendicular bisectors of a triangle are concurrent at a point equidistant
from the vertices of the triangle, known as the circumcenter
of the triangle.
The circumcenter is the center of a circle circumscribed in a triangle.
Some key facts:
• In an acute triangle, the circumcenter lies in the interior of the triangle.
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In a right triangle, the circumcenter lies on the hypotenuse.
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In an obtuse triangle, the circumcenter falls in the exterior of the triangle.
SUM IT UP!
Set of Lines in a Triangle Point of Concurrency Related Facts Medians Centroid Perpendicular Bisectors Circumcenter Angle Bisectors Incenter Altitudes Orthocenter The centroid divides each median into a 2 to 1 ratio. The distance from the vertex to the centroid is two‐thirds the distance of the entire median. The circumcenter is equidistant from the VERTICES of the triangle. We can CIRCUMSCRIBE a circle about the triangle. The center is at the circumcenter, and the length of the radius is the distance from the circumcenter to a vertex. The incenter is equidistant from the SIDES of the triangle when it is perpendicular to the sides. We can INSCRIBE a circle in the triangle. The center is at the incenter, and the length of the radius is the distance from the incenter to the side of the triangle where it intersects at a right angle. In an acute triangle, the orthocenter lies inside the triangle. In a right triangle, the orthocenter lies at the vertex of the right angle. In an obtuse triangle, the orthocenter lies outside the triangle.