Geometry CC 1.14 – Points of concurrency
Complete the table below
Point of
Type of
concurrency
segments
forming the
point
Properties
Centroid
Circumcenter
Incenter
Orthocenter
Exercise: Construct a centroid in the obtuse triangle below.
1.
Location for
acute triangle
(inside or
outside of
triangle)
Location for
obtuse triangle
(inside or
outside of
triangle)
2.
3.
4.
Geometry CC WS 1.14B Points of concurrency continued
Circumscribed circle (triangle is inscribed) – all vertices of a polygon are points on a circle
Center of the circle is the circumcenter.
Inscribed circle – each side of the polygon is tangent to the circle (intersects at one point)
Center of the circle is the incenter.
Exercise#1: Construct the circumcenter in the obtuse triangle below.
Exercise #2: Construct an inscribed circle in the triangle below.
3. Three medians of a triangle intersect at a point. Which measurements could represent the segment lengths of
a median?
(1) 2 and 3
(2) 3 and 4.5
(3) 3 and 6
(4) 3 and 9
4. Describe the geometric principle used in the construction below. Your description should include, the type of
segment, point of concurrency and circle used in the construction.
5. In the diagram below of ∆𝐴𝐵𝐶, ̅̅̅̅
𝐶𝐷 is the bisector of ∠𝐵𝐶𝐴, ̅̅̅̅
𝐴𝐸 is the bisector of ∠𝐶𝐴𝐵 and ̅̅̅̅
𝐵𝐺 is drawn.
Which statement must be true?
(1) 𝐷𝐺 = 𝐸𝐺
(2) 𝐴𝐺 = 𝐵𝐺
(3) ∠𝐴𝐸𝐵 ≅ ∠𝐴𝐸𝐶
6. In which triangle do the three altitudes intersect outside the triangle?
(1) a right triangle
(2) an acute triangle (3) an obtuse triangle
(4) ∠𝐷𝐵𝐺 ≅ ∠𝐸𝐵𝐺
(4) an equilateral triangle
7. For a triangle, which two points of concurrence could be located outside the triangle?
(1) incenter and centroid
(2) centroid and orthocenter
(3) incenter and circumcenter
(4) circumcenter and othocenter
8. In what type of triangle would the point of intersection of the three medians be the same as the point of
intersection of the three altitudes.
(1) scalene triangle (2) isosceles triangle
(3) equilateral triangle
(4) right isosceles triangle
9. State the difference between an inscribed circle and a circumscribed circle (include the point of concurrency
needed as the center point for each).