Let ABC be a triangle.Draw median AD on BC.Draw median AE of triangle ABD on BD.Take a point on side AB say F and join CF. Now from D make a line parallel to CF such that the line cuts AB at G. Note- G lies between F and B. AD intersect CF and P. AE intersect GD at H. So BD=DC Given AP=PD BE=ED CF parallel to GD. Find [AHD]/[ABC]

To find the ratio [AHD]/[ABC], we need to find the areas of triangles AHD and ABC.

1. First, let's find the area of triangle ABC.
The area of a triangle can be calculated using the formula: Area = (base * height) / 2.
In triangle ABC, BC is the base, and we need to find the height.

2. Median AD divides the base BC into two equal parts.
Therefore, BD = DC = BC/2.

3. Since AE is a median of triangle ABD, it divides BD into two equal parts.
So, BE = ED = BD/2.

4. From the given information, we know that CF is parallel to GD, and G lies between F and B.
This means that triangle CGD is similar to triangle CFB.

5. Since triangles CFB and CGD are similar, we can use the property of similar triangles to determine their corresponding side lengths.
The ratio of the lengths of corresponding sides in similar triangles is equal.

CF / CG = FB / GD
CF / CG = FB / (BE + ED) (Since GD = BE + ED)

Therefore, CF / CG = FB / (BD/2 + BD/2)
= FB / BD
= FB / (BC/2)

This implies CF / CG = 2 * FB / BC

6. Now, we know that AP is a median of triangle ABC, and AD intersects CF at point P.
The property of medians states that the point of intersection of medians divides the median into segments with a 2:1 ratio.
Therefore, PD = 2 * AP.

7. Now, we have all the required lengths to find the area of triangle AHD.
The height of triangle AHD is given by HD, and it can be calculated as:
HD = AE - HE.
We know that AW is a median of triangle ABD, so AE = 2 * AW.
Also, HE = (2/3) * HD. (Again, using the property of medians)

8. We can substitute the values of AE and HE into the equation HD = AE - HE to find HD (height of triangle AHD).

9. Finally, we can calculate the areas of triangles AHD and ABC:
Area of triangle AHD = (base * height) / 2 = (AD * HD) / 2
Area of triangle ABC = (base * height) / 2 = (BC * HD) / 2

10. With the areas of triangles AHD and ABC, we can calculate the ratio [AHD]/[ABC] by dividing the area of AHD by the area of ABC.