In rectangle ABCD, the bisector of angle C intersects side AB at point E, and the continuation

In rectangle ABCD, the bisector of angle C intersects side AB at point E, and the continuation of DA at point F, find the area of the rectangle if CE = 4, CF = 6.

Consider a right-angled triangle CDF (angle D is a straight line), it is isosceles, since (CF – bisector by condition). In this triangle, we know the hypotenuse CF = 6. Let us denote the length of the leg (FD and CD) x, and find it using the Pythagorean theorem:
2x² = 6²
x² = 18
x = √18 = 3√2
Consider triangles FEA and FCD – they are similar in two angles. Let’s write down the ratio of the parties:
FA / FD = FE / FC
FA = FE * FD / FC = 2 * 3√2 * 6 = √2
Now we can find AD.
AD = FD – FA = 3√2 – √2 = 2√2
Find the area of the rectangle:
S = AD * CD = 2√2 * 3√2 = 12
Answer: the area of the rectangle is 12.