In a triangle, if a line is drawn parallel to one side of the triangle, dividing the other two sides proportionally, what is this theorem called?

If two triangles have equal areas and one common side, which theorem can be used to confirm that they are between the same parallels?

In the context of parallelograms, what can be said about the areas of two parallelograms having the same base and height?

What is the relationship between the areas of two triangles on the same base and between the same parallels?

In a triangle, a median bisects both the triangle's area and the opposite side. What can be said about the triangle formed by the median?

In a triangle, if two medians intersect, what can be said about the ratio in which they divide each other?

In triangles with the same altitude, what determines the ratio of their areas?

Which theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length?

In a trapezium, if one pair of opposite sides are equal and parallel, what can be said about the trapezium?

If two triangles have equal areas and one vertex at the same point on a straight line, what is true about the line joining their other two vertices?

In a triangle ABC, if D is a point on side BC such that area of triangle ABD is equal to the area of triangle ACD, which theorem can be used to justify that AD is parallel to the base BC?

In triangle ABC, angle A is the right angle. If AB is perpendicular to BC, which theorem states that the area of triangle ABC can be calculated using the two legs AB and AC?

In a parallelogram ABCD, diagonal AC divides it into two triangles. Which theorem can be used to state that the areas of triangles ABC and ACD are equal?

In triangle ABC, if point D lies on side AB such that AD:DB = 2:1 and DE is drawn parallel to BC intersecting AC at E, what theorem helps us determine AE:EC?

In triangle ABC, AB = AC, and point D is on BC such that BD = DC. If AD is drawn, which theorem can be used to conclude that triangles ABD and ACD have equal areas?

In triangle ABC, if a line segment DE is drawn parallel to side BC, cutting sides AB at D and AC at E, what ratio does the Area of triangle ADE have with the Area of triangle ABC according to the Area Theorem?

In triangle PQR, if a line segment ST is drawn parallel to side QR and divides triangle PQR into two regions such that the area of triangle PST is half of the area of trapezoid STQR, what is the ratio of PS to PQ?

In triangle XYZ, line segment MN is drawn parallel to side XY, dividing the triangle into two smaller triangles and a trapezoid. If the area of triangle XMN is 1/4 of the area of triangle XYZ, what is the ratio of XM to XZ?

In triangle DEF, a line segment GH is drawn parallel to side EF. If GH divides the triangle such that the area of triangle DGH is 1/9 of the area of triangle DEF, what is the ratio of DG to DE?

In triangle JKL, line segment MN is drawn parallel to side KL, cutting sides JK and JL at points M and N, respectively. If JM:MK = 3:5, what is the area ratio of triangle JMN to triangle JKL?