Let's determine which statement is false. Each of the statements are referring to something being "concurrent." When three or more lines intersect at one point, they are concurrent. Let's look at each of the statements individually.
Statement A
Statement A says:
The bisectors of the angles
of a triangle are concurrent.
The intersection of the angle bisectors of a triangle creates the incenter of the triangle. All triangles have an incenter, so this statement is true.
Statement B
Statement B says:
The midsegments of a triangle are concurrent.
Midsegments of a triangle connect the midpoints of the sides of the triangles.
The midsegments do not all intersect at one point, they intersect each other one at a time. Therefore, they are not concurrent and this statement is false.
Statement C
Statement C says:
The perpendicular bisectors of the sides
of a triangle are concurrent.
The intersection of the perpendicular bisectors of the sides of a triangle creates the circumcenter of the triangle. Every triangle has a circumcenter, so this statement is true.
Statement D
Statement D says:
Four lines intersecting
in one point are concurrent.
This is the definition of concurrent. This statement is true.
