Concurrency HGEO

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Midsegment

Joins two midpoints of two sides of a triangle
Parallel and half the length of the 3rd side a triangle

Perpendicular Bisector

The bisector is also perpendicular
Equidistant (same distance) from the endpoints of the segments

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector

Angle Bisector

A ray, segment, or line, that divides an angle into two congruent angles

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the sides

Converse of Angle Bisector Theorem

If a point in the interior of a triangle is equidistant from the sides of the angle, then it is on the angle bisector

Median

A segment that connects a vertex to midpoint of the opposite side

Alititude

A perpendicular segment that goes from a vertex to the opposite side

Circumcenter

Perpendicular Bisector

Circumcenter Theorem

The circumcenter is equidistant to the vertices of the triangle

Circumcenter location

Acute triangle - inside
Obtuse - outside
Right - On
Same as orthocenter locations

Incenter

Angle Bisector

Incenter theorem

The incenter is equidistant to the sides of the triangle

Incenter/Centroid location

ALWAYS INSIDE

Incenter

Always "in the center", always inside the triangle

Centroid

Median

Centroid Theorem

The centroid is the balancing point of the triangle.
2/3 and 1/3 split of the median (longer one goes from the centroid to the vertex)

Centroid/Incenter location

ALWAYS INSIDE

Orthocenter

Altitude

Orthocenter location

Acute - Inside
Obtuse - Outside
Right - on
Same as circumcenter locations

Triangle Inequality Theorem

The sum of the lengths of any two sides must be greater than the third side (shortcut: add the two shortest lengths and compare to the other side)

Scalene Triangle Inequality

The longest side is opposite the largest angle

The point that is equidistant from the sides of ∆ABC is called the _____.

Incenter

The point that is equidistant from the vertices of ∆ABC is called the _____

Circumcenter

Centroid facts

The point that divides the median into a 2/3, 1/3 split is called the triangle’s _____

In what kind of triangle does the circumcenter lie on of the circle?

Right Triangle

Circumcenter

=dist. to vertices

Incenter

=dist. to sides

Centroid formula

((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)