Ptolemy’s theorem says that for a cyclic quadrilateral 𝐴𝐵𝐶𝐷, AC·BD = AB·CD + BC·AD.
With ruler and a compass, draw an example of a cyclic quadrilateral. Label its vertices 𝐴, 𝐵, 𝐶, and 𝐷.
Draw the two diagonals 𝐴𝐶 and 𝐵𝐷
With a ruler, test whether or not the claim that AC·BD = AB·CD + BC·AD seems to hold true.
Repeat for a second example of a cyclic quadrilateral.
Challenge: Draw a cyclic quadrilateral with one side of length zero. What shape is this cyclic quadrilateral?
Does Ptolemy’s claim hold true for it?
Exploratory Challenge: A Journey to Ptolemy’s Theorem
The diagram shows cyclic quadrilateral 𝐴𝐵𝐶𝐷 with diagonals 𝐴𝐶 and 𝐵𝐷
intersecting to form an acute angle with degree measure 𝑤. 𝐴𝐵 = 𝑎, 𝐵𝐶 = 𝑏, 𝐶𝐷 = 𝑐, and 𝐷𝐴 = 𝑑.
a. From the last lesson, what is the area of quadrilateral 𝐴𝐵𝐶𝐷 in terms of the lengths of its diagonals and the angle 𝑤? Remember this formula for later.
b. Explain why one of the angles, ∠𝐵𝐶𝐷 or ∠𝐵𝐴𝐷, has a measure less than or equal to 90°.
c. Let’s assume that ∠𝐵𝐶𝐷 in our diagram is the angle with a measure less than or equal to 90°. Call its measure 𝑣 degrees. What is the area of triangle 𝐵𝐶𝐷 in terms of 𝑏, 𝑐, and 𝑣? What is the area of triangle 𝐵𝐴𝐷 in terms of 𝑎, 𝑑, and 𝑣? What is the area of quadrilateral 𝐴𝐵𝐶𝐷 in terms of 𝑎, 𝑏, 𝑐, 𝑑, and 𝑣?
d. We now have two different expressions representing the area of the same cyclic quadrilateral 𝐴𝐵𝐶𝐷. Does it seem to you that we are close to a proof of Ptolemy’s claim?
e. Trace the circle and points 𝐴, 𝐵, 𝐶, and 𝐷 onto a sheet of patty paper. Reflect triangle 𝐴𝐵𝐶 about the perpendicular bisector of diagonal 𝐴𝐶. Let 𝐴′, 𝐵′, and 𝐶′ be the images of the points 𝐴, 𝐵, and 𝐶, respectively.
i. What does the reflection do with points 𝐴 and 𝐶?
ii. Is it correct to draw 𝐵′ as on the circle? Explain why or why not.
iii. Explain why quadrilateral 𝐴𝐵′𝐶𝐷 has the same area as quadrilateral 𝐴𝐵𝐶𝐷.
f. The diagram shows angles having degree measures 𝑢, 𝑤, 𝑥, 𝑦, and 𝑧.
Find and label any other angles having degree measures 𝑢, 𝑤, 𝑥, 𝑦, or 𝑧, and justify your answers.
g. Explain why 𝑤 = 𝑢 + 𝑧 in your diagram from part (f).
h. Identify angles of measures 𝑢, 𝑥, 𝑦, 𝑧, and 𝑤 in your diagram of the cyclic quadrilateral 𝐴𝐵′𝐶𝐷 from part (e).
i. Write a formula for the area of triangle 𝐵′𝐴𝐷 in terms of 𝑏, 𝑑, and 𝑤. Write a formula for the area of triangle 𝐵′𝐶𝐷 in terms of 𝑎, 𝑐, and 𝑤.
j. Based on the results of part (i), write a formula for the area of cyclic quadrilateral 𝐴𝐵𝐶𝐷 in terms of 𝑎, 𝑏, 𝑐, 𝑑, and 𝑤.
k. Going back to part (a), now establish Ptolemy’s theorem.
PTOLEMY’S THEOREM: For a cyclic quadrilateral 𝐴𝐵𝐶𝐷, AC·BD = AB·CD + BC·AD
CYCLIC QUADRILATERAL: A quadrilateral with all vertices lying on a circle is known as a cyclic quadrilateral
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