Diagonals of a trapezoid intersect each other at point E, and the extensions of the legs of the trapezoid intersect each other at point F. Prove that line EF bisects the bases of the trapezoid. PLEASE HELP, IT’S DUE TOMORROW. 50 POINTS AND BRAINLIEST.

See Below.

Step-by-step explanation:

Let the intersection point above E be K, and let the intersection point below E be J.

Since ABCD is a trapezoid:

Therefore:

And:

So, by AA-Similarity:

CPSTP (Corresponding Parts of Similar Triangles are Proportional):

Likewise, do the same to the other side. Since AB is parallel to DC:

Hence:

By CPSTP:

Substitution:

Since AB is parallel to DC:

By vertical angles:

Hence, by AA-Similarity:

CPSTP:

Likewise:

And by vertical angles:

Hence:

CPSTP:

Substitution:

Now, we have obtained that:

And:

In the second proportion, multiply both sides by JA:

In the first equation, multiply both sides by JB:

Substitute:

Cross-multiply:

Cancel out KD and simplify:

Therefore:

And using the first proportion again:

Hence, EF bisects AB and DC, the bases of the trapezoid.