Centers of a Triangle: Centroid, Incenter, Circumcenter, Orthocenter

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If the area of triangle ABC is 108 sq. cm, then the area of triangle EOD, in sq. cm, is:

In a triangle ABC, medians AD and BE are perpendicular to each other and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq. cm, is:

From a triangle ABC with sides 40 ft, 25 ft, and 35 ft, a triangular portion GBC is cut off where G is the centroid. The area, in sq. ft, of the remaining portion of triangle ABC is:

In ΔABC, AB = 7 cm, AC = 9 cm, and BC = 8 cm. Find the length of the median AD drawn to side BC.

In ΔABC, medians BE and CF are mutually perpendicular to each other and intersect at G. If AB = 19 cm and AC = 22 cm, what is the length of side BC?

The three medians of a triangle have lengths 9 cm, 12 cm, and 15 cm. Find the area of the triangle.

In a right-angled triangle ΔABC with legs AB = 6 cm, BC = 8 cm, and hypotenuse CA = 10 cm, find the distance from vertex A to the centroid G.

The sum of the squares of the sides of a triangle is 240 cm². What is the sum of the squares of its three medians?

In ΔABC, the three medians intersect at centroid G. If the area of ΔABC is 108 cm², what is the area of quadrilateral BDGF (where D, F are midpoints of BC, AB)?

In ΔABC, G is the centroid. A line passing through G and parallel to BC intersects AB at P and AC at Q. If the area of ΔABC is 90 cm², find the area of ΔAPQ.