Question 11.4.4

Let’s see what happens when we apply Ptolemy’s theorem (see Question 10.6.3) to a cyclic quadrilateral with one of its diagonals forming a diameter of the circle. Diagram: ptolemy 4

  1. Given that $AC$ is a diameter of the circle through $A$, $B$, $C$, and $D$, find $\angle ABC$ and $\angle ADC$. Show answer

  2. Find $BD$ in terms of $\theta$, $\phi$, and the diameter, $AC$, where $\theta = \angle ACB$ and $\phi = \angle ACD$. Show hint

    Show answer

  3. Now apply Ptolemy’s theorem, \[AC\cdot BD = AB\cdot CD + BC\cdot AD.\] Show hint

    Show answer