Invariant Product related to Triangles on the sides of a Triangle
plus Anghel's Hexagon Concurrency theorem

Arbitrary triangles ABD, BCE and CAF have been constructed on the sides of △ABC in the dynamic sketch below. Sides and angles have been measured and respective products calculated as shown.

Explore
1) Click on the 'Show Product' button.
2) What you notice about the product of $$\frac{AD}{DB}\times \frac{BE}{EC}\times \frac{CF}{FA}\times \frac{sinDAB}{sinDBA}\times \frac{sinEBC}{sinECB}\times \frac{sinFCA}{sinFAC}?$$
3) Is it always true? Make a conjecture and explore by dragging any of the red vertices.

Invariant Product & Anghel's Hexagon Concurrency theorem

Challenge
4) Can you explain why (prove) your observation in 3) is true?
Hint: Use the sine rule.

Further Generalization
5) From your explanation/proof in 4), can you generalize this invariant product further to quadrilaterals, pentagons, etc.?

Published Paper
Read my paper The Sine Rule Disguised in the Dec 2024 issue of the journal Learning and Teaching Mathematics, No. 37, pp. 36-37. The journal is published by the Association for Mathematics Education of South Africa (AMESA) - all rights reserved.

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Explore More
6) Click on the 'Show Lines' button.
7) What do you notice about the red lines? Are they always concurrent?
8) Can you find necessary and sufficient conditions for the red lines to be concurrent? Explore dynamically!
9) Check your answer to 8), by clicking on the 'Show Anghel's Hexagon Concurrency theorem' button to navigate to a new sketch.
10) In the new sketch, click on the 'Show Lines' button.
11) What do you notice? Check your findings by dragging. Can you formulate a conjecture?

Anghel's Hexagon Concurrency theorem
Given a hexagon ADBECF, then its main diagonals AE, BF and CD are concurrent (see Anghel, 2016, 2018), if and only if, $$\frac{sinDAB}{sinDBA}\times \frac{sinEBC}{sinECB}\times \frac{sinFCA}{sinFAC}\times \frac{sinDBC}{sinABE}\times \frac{sinECA}{sinBCF}\times \frac{sinFAB}{sinCAD}=1$$
Or equivalently from 2) and 4), the main diagonals are concurrent, if and only if, $$\frac{AD}{DB}\times \frac{BE}{EC}\times \frac{CF}{FA}=\frac{sinDBC}{sinABE}\times \frac{sinECA}{sinBCF}\times \frac{sinFAB}{sinCAD}$$
Note that this theorem generalizes Jacobi's Theorem, and as shown by Anghel (2018), this theorem can be viewed as equivalent to a generalization of the trigonometric version of Ceva's theorem.

Latest Update
On 20 Jan 2025, in the Facebook Group 'Romantics of Geometry', Kousik Sett from India posted the following straight forward, alternative proof of Anghel's theorem.

References
Anghel, N. (2016). Concurrency and Collinearity in Hexagons. Journal for Geometry and Graphics, Volume 20, No. 2, 159–171.
Anghel, N. (2018). Concurrency in Hexagons - a Trigonometric Tale. Journal for Geometry and Graphics, Volume 22, No. 1, 21–29.
Sett, K. (2025). Solution by Kousik Sett. Romantics of Geometry (Ρομαντικοί της Γεωμετρίας).

Some Applications of Anghel's theorem
Concurrency, collinearity and other properties of a particular hexagon
Jacobi's Generalization of the Fermat-Torricelli point
An extension of the IMO 2014 Problem 4
Another concurrency related to the Fermat point of a triangle

Related Links
Concurrency, collinearity and other properties of a particular hexagon
Jacobi's Generalization of the Fermat-Torricelli point
An extension of the IMO 2014 Problem 4
De Villiers points of a triangle
Another concurrency related to the Fermat point of a triangle
Power Lines of a Triangle
Haag Hexagon and its generalization to a Haag Polygon
Haag Hexagon - Extra Properties
Easy Hexagon Explorations
Some Circle Concurrency Theorems
Cosine-Sine Angle Rule

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Free Download of Geometer's Sketchpad

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