Proving Centers of Triangles Adapted from Walch Education
Slide 3
Circumcenter The perpendicular bisector is the line that is
constructed through the midpoint of a segment. The three
perpendicular bisectors of a triangle are concurrent, or intersect
at one point. This point of concurrency is called the circumcenter
of the triangle. The circumcenter of a triangle is equidistant, or
the same distance, from the vertices of the triangle. This is known
as the Circumcenter Theorem. 1.9.4: Proving Centers of
Triangles2
Slide 4
3 Theorem Circumcenter Theorem The circumcenter of a triangle
is equidistant from the vertices of a triangle. The circumcenter of
this triangle is at X.
Slide 5
Circumcenter, continued The circumcenter can be inside the
triangle, outside the triangle, or even on the triangle depending
on the type of triangle. The circumcenter is inside acute
triangles, outside obtuse triangles, and on the midpoint of the
hypotenuse of right triangles. 1.9.4: Proving Centers of
Triangles4
Slide 6
Circumcenter, continued 1.9.4: Proving Centers of Triangles5
Acute triangleObtuse triangleRight triangle X is inside the
triangle. X is outside the triangle. X is on the midpoint of the
hypotenuse.
Slide 7
Circumcenter, continued The circumcenter of a triangle is also
the center of the circle that connects each of the vertices of a
triangle. This is known as the circle that circumscribes the
triangle. 1.9.4: Proving Centers of Triangles6
Slide 8
Incenter The angle bisectors of a triangle are rays that cut
the measure of each vertex in half. The three angle bisectors of a
triangle are also concurrent. This point of concurrency is called
the incenter of the triangle. The incenter of a triangle is
equidistant from the sides of the triangle. This is known as the
Incenter Theorem. 1.9.4: Proving Centers of Triangles7
Slide 9
8 Theorem Incenter Theorem The incenter of a triangle is
equidistant from the sides of a triangle. The incenter of this
triangle is at X.
Slide 10
Incenter, continued The incenter is always inside the triangle.
1.9.4: Proving Centers of Triangles9 Acute triangleObtuse
triangleRight triangle
Slide 11
Incenter, continued The incenter of a triangle is the center of
the circle that connects each of the sides of a triangle. This is
known as the circle that inscribes the triangle. 1.9.4: Proving
Centers of Triangles10
Slide 12
Orthocenter The altitudes of a triangle are the perpendicular
lines from each vertex of the triangle to its opposite side, also
called the height of the triangle. The three altitudes of a
triangle are also concurrent. This point of concurrency is called
the orthocenter of the triangle. 1.9.4: Proving Centers of
Triangles11
Slide 13
Orthocenter, continued The orthocenter can be inside the
triangle, outside the triangle, or even on the triangle depending
on the type of triangle. The orthocenter is inside acute triangles,
outside obtuse triangles, and at the vertex of the right angle of
right triangles. 1.9.4: Proving Centers of Triangles12
Slide 14
Orthocenter, continued 1.9.4: Proving Centers of Triangles13
Acute triangleObtuse triangleRight triangle X is inside the
triangle. X is outside the triangle. X is at the vertex of the
right angle.
Slide 15
Centroid The medians of a triangle are segments that join the
vertices of the triangle to the midpoint of the opposite sides.
Every triangle has three medians. The three medians of a triangle
are also concurrent. This point of concurrency is called the
centroid of the triangle. The centroid is always located inside the
triangle the distance from each vertex to the midpoint of the
opposite side. This is known as the Centroid Theorem. 1.9.4:
Proving Centers of Triangles14
Slide 16
1.9.4: Proving Centers of Triangles15 Theorem Centroid Theorem
The centroid of a triangle is the distance from each vertex to the
midpoint of the opposite side. The centroid of this triangle is at
point X.
Slide 17
Centroid, continued The centroid is always located inside the
triangle. The centroid is also called the center of gravity of a
triangle because the triangle will always balance at this point.
1.9.4: Proving Centers of Triangles16 Acute triangleObtuse
triangleRight triangle
Slide 18
Point of Concurrency Each center serves its own purpose in
design, planning, and construction. 1.9.4: Proving Centers of
Triangles17 Center of triangleIntersection of
CircumcenterPerpendicular bisectors IncenterAngle bisectors
OrthocenterAltitudes CentroidMedians