Explanation of a proof without words of Ptolemy theorem

Question

What is the explanation of Ptolemy Theorem - Proof Without Words?

Answer 1

If you look at $\triangle BCD$ in the left part of the picture, you see that

• the length of the side opposite angle $\alpha$ is $a$;
• the length of the side opposite angle $\beta$ is $b$;
• the length of the side opposite angle $\gamma+\delta$ is $f$.

In the right part of the picture, the triangle labeled $a\cdot(\triangle BCD)$ has the same three angles, and the same three lengths of opposite sides except that they're multiplied by $a$.

Do the same with $\triangle ABC$, which also appears in the left part, but this time multiply all three sides by $f$, getting $f\cdot(\triangle ABC)$, which appears in the right part. That triangle has a side of length $af$, and the one I mentioned in the paragraph above also has a side of length $af$, so those two fit together, as shown in the right part.

And similarly with $b\cdot(\triangle BAD)$.

Because the four angles between sides and diagonals have to add up to $180^\circ$, i.e. $\alpha+\beta+\gamma+\delta=180^\circ$, you get a straight line segment that is the union of two segments of lengths $ac$ and $bd$, the two products of the lengths of pairs of opposite sides. But the lower line segment of the parallelogram has length $ef$, so we have $ac+bd=ef$, which is what Ptolemy's theorem says.

But how do we know it's a parallelogram? For that it suffices to look at the angles in the four extreme corners: $\alpha+\delta$ appears in each of two opposite corners, and $\beta+\gamma$ in the other two.

Ptolemy, in the second century AD, used this theorem in the construction of Ptolemy's table of chords.

(I thought that I knew that Ptolemy lived in Alexandria, just as Euclid and Eratosthenes did, and just as Archimedes lived in Syracuse. So I was a bit surprised when I read in Nicholas Nicastro's book Circumference that Ptolemy's case is unlike those of Euclid, Eratosthenes, and Archimedes, in this respect: No ancient writings by Ptolemy or others say where he lived, but a very strong circumstantial case can be made that it was Alexandria. Nicastro was for a time a professor of history, and here we see one of the differences between the way historians think and the way mathematicians think. The latter say casually that Ptolemy lived in Alexandria and don't think about where that information came from.)

Answer 2

Nice proof!

The idea is that all the angles shown are equal to the angles in the picture on the left. This implies that the figure on the right is a parallelogram. Therefore, the two opposite sides are equal.

Is it clear why the angles match? It's based on the congruence of corresponding triangles.

Answer 3

The authors W. Derrick and J. Herstein wisely dilate the triangles BCD, BAD and ABC respectively by factors a, b and f (without changing their interior angles) to form three new similar triangles. With their equal common sides, the 3 triangles form a quadrilateral having two pairs of opposite angles. Therefore the quadrilateral so formed is a parallelogram, in which the opposite sides are equal and hence ef=ac+bd.