Question bank: Triangles: Cevians and Notable Points

In a triangle ABC, a cevian is a line segment that connects a vertex of the triangle to any point on the support line of the opposite side. Let P be a point of intersection of the internal angle bisectors of triangle ABC. What is the name given to point P in relation to the cevians of triangle ABC? Justify your answer based on the properties of the notable points of the triangle and the internal angle bisectors.

A fictional city called Poligolândia is organized in the shape of an isosceles triangle. In the center of this city, there is a circular building that also serves as the government headquarters. From this center, the city's rulers decided to build three cevians that connect each side of the triangle to the central point of the circular building. The cevians divide the triangle into three parts, and the perimeter of the city is 90 km.

Which of the cevians divides one of the angles of the triangle into two equal regions?

In a dodgeball game in Physical Education class, the students decided to mark the playing field in the shape of a triangle in the school yard. One of the students drew a cevian connecting vertex A to the opposite side measuring 18 meters. This cevian divides the opposite side into two segments with lengths x and 6. Determine the value of x and the ratio between the sides divided by the cevian.

In a triangle ABC, consider the cevian AP that extends through the midpoint P of side BC. Also consider the cevians BQ and CR that pass through the midpoints of AC and AB, respectively, and meet at P.