Ptolemy on Hinges
Ptolemy's theorem tells us that in a convex cyclic quadrilateral ABPC the sum of the products of the two pairs of opposite sides equals the product of its two diagonals. In other words,
AC×BP + AB×CP = AP×BC
|What if applet does not run?|
Rotate triangle ACP around A through the angle CAB so that C is mapped onto a point, say C', on AB. P is mapped to P'.
In case ΔABC is equilateral, one may perceive the semblance with the configuration from a proof of van Schooten's theorem. When ΔABC is equilateral the rotation places P' on the extension of BP. The same is true when ΔABC is isosceles. In the general case, we may only claim that C'P'||BP. (The reason is the same in all three cases. Since the quadrilateral ABPC is cyclic
Triangles ABP'' and AC'P' are similar, implying
AC' / C'P' = AB / BP'',
|BP''||= AB×C'P' / AC'|
|= AB×CP / AC.|
Triangles ABC and AP''P are also similar because, in addition to
AC / AP = BC / PP''
PP'' = BC×AP / AC
But PP'' = BP + BP'' = BP + AB×CP / AC. We see that
BP + AB×CP / AC = BC×AP / AC.
Multiplying by AC gives Ptolemy's identity.
- A. Percy and D. G. Rogers, Alternative route: from van Schooten to Ptolemy, Normat 57:3, 116-128 (2009)
- Ptolemy's Theorem
- Sine, Cosine, and Ptolemy's Theorem
- Useful Identities Among Complex Numbers
- Ptolemy on Hinges
- Thébault's Problem III
- Van Schooten's and Pompeiu's Theorems
- Ptolemy by Inversion
- Brahmagupta-Mahavira Identities
- Casey's Theorem
- Three Points Casey's Theorem
- Ptolemy via Cross-Ratio
- Ptolemy Theorem - Proof Without Word
- Carnot's Theorem from Ptolemy's Theorem
Copyright © 1996-2018 Alexander Bogomolny