Download Point of Concurrency

3.1 Special Segments and Centers of Triangles
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Define and recognize perpendicular bisectors, angle bisectors, medians, and altitudes. Define and recognize points of concurrency.
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Classifications of Triangles:
By angle 1. Acute: A triangle with three acute angles.
Classifications of Triangles:
By Side:
1. Equilateral: A triangle with three congruent sides.
2. Isosceles: A triangle with at least two congruent sides.
3. Scalene: A triangle with three sides having different lengths. (no sides are congruent)
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Special Segments and Centers in Triangles
A Perpendicular Bisector is a segment or line that passes through the midpoint of a side and is perpendicular to that side.
2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle
4. Equiangular: A triangle with three congruent angles
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Jul 24­9:36 AM
Point of Concurrency
The three perpendicular bisectors of a triangle intersect at a single point.
Two lines intersect at a point. When three or more lines intersect at the same point, it is called a "Point of Concurrency."
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The point of concurrency of the perpendicular bisectors is called the circumcenter.
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Circumcenter Properties
1. The circumcenter is the center of the circumscribed circle.
An angle bisector is a segment that divides an angle into two congruent angles. BD is an angle bisector.
2. The circumcenter is equidistant to each of the triangles vertices.
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The three angle bisectors of a triangle intersect at a single point.
The point of concurrency of the angle bisectors is called the incenter.
Point A is the incenter of the triangle
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An altitude is a segment from a vertex perpendicular to the opposite side
m∠ABD= m∠DBC
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Incenter properties
1. The incenter is the center of the inscribed circle
2. The incenter is equidistant to each side of the triangle.
AB = AD = AC
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The three altitudes of a triangle are concurrent. The point of concurrency is called the orthocenter.
m∠ADB= m∠ADC=90°
Point A is the orthocenter
of the triangle
AD is an altitude of ∆ABC
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Jul 24­9:36 AM
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A median is a segment from a vertex
to the midpoint of the opposite side
The three medians of the triangle are concurrent. The point of concurrency is called the centroid.
AB is a median of ∆ACD
Point A is the centroid of the triangle.
Jul 24­9:36 AM
Jul 24­9:36 AM
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