Triangle Centers
What point is formed by the intersection of medians?
Centroid
What does “concurrent” mean?
Lines intersect at one point
What must opposite angles equal in a cyclic quadrilateral?
180°
What value must the product equal for concurrency?
1
True or False: All triangles have an incircle.
True
Which center is equidistant from the sides of a triangle?
Incenter
What is a cevian?
Segment from vertex to opposite side
If opposite angles are not supplementary, what can you conclude?
Not cyclic
If the product ≠ 1, what does that mean?
Cevians are not concurrent
True or False: The centroid can lie outside a triangle.
False - Why?
Which center is formed by altitudes?
Orthocenter
What is a cyclic quadrilateral?
A quadrilateral whose vertices lie on a circle
A quadrilateral has angles 100°, 80°, 95°, 85°. Is it cyclic?
Yes (100+80=180, 95+85=180)
If AX/XB = 2, BY/YC = 3, CZ/ZA = 1/6 → are they concurrent?
Yes (product = 1)
True or False: All cevians are concurrent.
False - Why?
Which center is equidistant from the vertices?
Circumcenter
What is an incircle?
Circle tangent to all sides of a triangle
Why is a rectangle always cyclic?
Opposite angles are 90° + 90° = 180°
If AX/XB = 2, BY/YC = 2, CZ/ZA = 1/2 → concurrent?
No (product = 2)
A triangle has perpendicular bisectors that meet outside. What type of triangle?
What ratio does the centroid divide each median?
2:1
What is a circumcircle?
Circle passing through all vertices
What is another way to describe a cyclic quadrilateral?
All vertices lie on a circle
What theorem is used to determine if cevians are concurrent?
Ceva’s Theorem
Name ALL four triangle centers.
Centroid, Orthocenter, Incenter, Circumcenter