Let's break down and work through the problem step by step based on the diagram and the questions given.

Question 10.1:

You're asked to prove three things:

10.1.1: Prove that $\angle C = \angle ABF$

• Since $BF$ is a tangent to the circle at $B$ and $D$ is the center of the circle, we can use the tangent-chord theorem. This theorem states that the angle between a tangent and a chord (such as $\angle ABF$) is equal to the angle subtended by the chord on the opposite side of the circle (which in this case is $\angle C$).
• Hence, $\angle C = \angle ABF$.

10.1.2: Prove that $DF \cdot BC = AB \cdot BD$

• By applying the power of a point theorem (also known as the intersecting chords theorem), we know that for two intersecting chords in a circle, the products of the segments of each chord are equal.
• Here, $DF \cdot BC = AB \cdot BD$ because $D$, $B$, $F$, and $C$ form intersecting chords inside the circle.

10.1.3: Prove that $E$ is the midpoint of the circle that passes through $D$, $B$, and $F$

• $BE = DE$ is given in the problem.
• Since $BE = DE$, $E$ lies on the perpendicular bisector of $BD$, and it is the midpoint of the line that connects $D$, $B$, and $F$.
• Additionally, because $BE = DE$, $E$ must also be equidistant from points $D$, $B$, and $F$, implying that it is the center of the circumcircle passing through these three points.

Question 10.2:

You are asked to prove the equation:

$\frac{1}{BC^2 - AB^2} = \frac{AD \cdot FB}{EF \cdot AC^3}$

This equation involves geometric relationships between the sides and segments of the circle. To prove this, we need to:

• Use properties of similar triangles, particularly between the triangles formed by these line segments (such as $\triangle ADB$ and $\triangle EBF$).
• Apply proportionality rules between the sides, using the fact that similar triangles have proportional corresponding sides.
• Utilize any given geometric relationships in the diagram (such as the fact that $BF$ is a tangent) to set up ratios that can lead to the desired proof.

Would you like me to go into more detail on each step or assist with specific parts of these proofs?

Follow-up Questions:

1. Do you need further explanation on how to apply the tangent-chord theorem?
2. Would you like help proving the intersecting chords theorem used in 10.1.2?
3. Should we explore the properties of similar triangles for solving 10.2 in more depth?
4. Do you need a review of how to use the power of a point theorem in circles?
5. Would you like a detailed explanation of how to prove that $E$ is the midpoint in 10.1.3?

Tip:

When working with circle theorems, always look for tangents, chords, and angles subtended by arcs—they often lead to simple but powerful geometric relationships.