Problem Analysis:

The problem contains a circle $ABCD$ with the following given details:

1. $BD$ is the diameter of the circle.
2. $E$ is a point on $BD$ such that $AE \parallel DC$.
3. $AC$ and $BD$ intersect at $F$.
4. $AF = FC$.
5. $\angle ACB = 61^\circ$.

The questions are:

1. (a) Find $\angle DBC$.
2. (b) Determine whether $E$ is the center of the circle and explain your answer.

Solution for (a): Find $\angle DBC$

Key observations:

1. Since $BD$ is the diameter, $\angle BAD = 90^\circ$ (an angle subtended by the diameter of a circle at the circumference is a right angle).
2. $AF = FC$, so $F$ is the midpoint of chord $AC$. Thus, $\triangle AFC$ is isosceles.
3. The given $\angle ACB = 61^\circ$.

Step-by-step calculation:

• In $\triangle ABC$, the sum of the angles is $180^\circ$: $\angle BAC + \angle ACB + \angle ABC = 180^\circ.$ Substituting $\angle ACB = 61^\circ$: $\angle BAC + \angle ABC = 119^\circ.$

• Since $\triangle ABC$ is isosceles ($AF = FC$), $\angle BAC = \angle ABC$. Let $\angle BAC = \angle ABC = x$. Then: $x + x = 119^\circ \implies 2x = 119^\circ \implies x = 59.5^\circ.$ Thus: $\angle BAC = \angle ABC = 59.5^\circ.$

• Now, consider $\triangle DBC$. Since $\angle DBC$ subtends arc $DC$ and $\angle ABC$ subtends the same arc, the angles are equal: $\angle DBC = \angle ABC = 59.5^\circ.$

Final Answer for (a):

$\boxed{\angle DBC = 59.5^\circ.}$

Solution for (b): Is $E$ the center of the circle?

Key observations:

1. $E$ lies on the diameter $BD$.
2. For $E$ to be the center of the circle, it must be equidistant from all points on the circumference. This is equivalent to $AE = BE = DE = CE$.

Reasoning:

• From the problem, we know that $AE \parallel DC$. This implies that $E$ is not the midpoint of $BD$ (since no information suggests that $AE$ is perpendicular to $BD$, which is necessary for $E$ to be the center).
• Furthermore, the given data does not provide any evidence that $E$ satisfies the property of being equidistant from all points on the circle.

Conclusion:

$E$ is not the center of the circle.

Final Answer for (b):

\boxed{\text{E is not the center of the circle.}}

Do you have any questions or would you like further explanation?

5 Related Questions:

1. What is the property of an angle subtended by a diameter in a circle?
2. How do you prove that a point is the center of a circle using geometry?
3. What happens to the angles in an isosceles triangle when one angle is given?
4. How does parallelism ($AE \parallel DC$) influence the geometry of the problem?
5. Can $E$ ever be the center of the circle if it lies on the diameter?

One Tip:

When solving geometry problems in circles, always leverage key properties such as the angle subtended by a diameter being $90^\circ$ and the relationship between angles subtending the same arc.