Prove that triangle ADK is similar to triangle FEK, and AK * KE = DK * KF.

Solution:

• Part 1: Prove that triangle ADK is similar to triangle FEK.
• We need to find two pairs of equal angles.
• Angle AKD and angle FKE are vertically opposite angles, so $$\angle AKD = \angle FKE$$.
• Angle DAK and angle FEK are inscribed angles subtended by the same arc DK. Therefore, $$\angle DAK = \angle FEK$$.
• Since two angles of triangle ADK are equal to two angles of triangle FEK, the triangles are similar by the AA similarity criterion.
• Thus, $$\triangle ADK \sim \triangle FEK$$.
• Part 2: Prove that AK * KE = DK * KF.
• From the similarity of the triangles $$\triangle ADK \sim \triangle FEK$$, we have the ratio of corresponding sides equal:
• $$\frac{AK}{FK} = \frac{DK}{EK} = \frac{AD}{FE}$$
• From the first part of the ratio, $$\frac{AK}{FK} = \frac{DK}{EK}$$.
• Cross-multiplying gives: $$AK \cdot EK = DK \cdot FK$$.

Answer: The triangles ADK and FEK are similar by AA similarity, which leads to the proportion $$\frac{AK}{FK} = \frac{DK}{EK}$$. Cross-multiplying yields AK \cdot KE = DK \cdot KF.