Light Ray in a Triangle (Fagnano's Minimal Path)

The dynamic geometry activities below are from the "Proof as Verification" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Light Ray in a Triangle Worksheet & Teacher Notes.

Prerequisites
Knowledge of the concurrency of the altitudes of a triangle and of the properties of cyclic quadrilaterals (e.g., that equal angles are subtended on same chord; opposite angles supplementary implies that a convex quadrilateral is cyclic). Specifically, it is recommended that students have already completed the following online activities:
Water Supply: Four Towns (Perpendicular Bisectors)
Cyclic Quadrilateral
Cyclic Quadrilateral Converse
Triangle Altitudes (Concurrency)

Light Ray in a Triangle (Fagnano's Minimal Path)

Notes
1) Users are strongly encouraged to download the accompanying Worksheet, and to print it out to use in conjunction with the dynamic sketch above. Alternatively, copy the URL: http://dynamicmathematicslearning.com/light-ray-rethinking-proof.pdf and paste it into a new browser window. Then resize the new window to place it side by side with this one, or one below the other. (However, this is not likely to be a feasible option for users using small screens such as a cellphone or tablet.)
2) Click on 'Send light ray' button, and watch its path traced out as it reflects in the mirrors BC and CA.
(Note: this trace path will remain in the sketch, but you can remove it by reloading the webpage in your browser.)
3) Note that the perimeter of triangle XYZ is given. Drag point X until the perimeter of triangle XYZ is minimized.
4) Once you've found a location where the perimeter of XYZ is minimized, click on the 'Show altitudes and feet' button. What do you notice?
5) Click on the 'Show sides of △XYZ' and 'Show angles at ...' buttons. What do you notice? Check your observations by dragging.
6) Change the shape of ABC by dragging (but keeping it acute), and repeat steps 3) to 5).
7) What do you notice? What conjectures can you formulate?
8) Challenge: Can you explain why (prove that) your observations in 7) are always true?
(Some proof constructions)
a) Use the Point tool on the left to construct the point of concurrency of the altitudes (say O).
b) Use the Arc tool on the left (scroll down if necessary) to construct a circle around OECF.
c) Use the Reflect tool on the left (scroll down if necessary) to reflect point Y in sides AB and AC.
d) Use the Segment tool on the left (scroll down if necessary) to draw segments Y'₁X and Y'₂Z.
(Unfortunately it is not possible with WebSketchpad, unlike the free desktop version of Sketchpad, to label new geometric elements in a webpage.)
9) For a proof of the uniqueness of the minimal perimeter solution, consult Hildebrandt & Tromba (1985, pp. 60– 63), Courant & Robbins (1996, pp. 346-352) or Coxeter & Greitzer (1967).
10) An interesting alternative problem setting and geometric solution with physics is provided by Rane (2017). Useful historical background and pedagogical ideas are provided in Adesso et al (2022).

Historical Background
Fagnano's problem, posed by Giovanni Fagnano dei Toschi in 1775, asks for the inscribed triangle with the minimum perimeter inside a given acute-angled triangle. The solution, found by Fagnano using calculus, and later proven geometrically by Hermann Schwarz (1843–1921) and others, is the triangle formed by the feet of the altitudes of the original triangle, known as the orthic triangle. This problem is also related to billiard physics, as the orthic triangle represents the shortest periodic path (like a billiard ball's) within the triangle.

References
Adesso, M. G.; Capone, R.; Fiore, O. (2022). The Historical Fagnano's Problem: teaching materials as artifacts to experiment mathematical and physical Tasks in Italian High School. Paper in Barbin, E.; Capone, R.; Fried, M.N.; Menghini, M.; Pinto, H.; Tortoriello, S. (editors). History and Epistemology in Mathematics Education Proceedings of the 9th European Summer University, 18‐22 July 2022, pp. 614-224.
Courant, R., Robbins, H. & Stewart, I. (1996, 2nd Edition). Schwarz's Triangle Problem, pp. 21–29. What is Mathematics?, Oxford University Press.
Coxeter, H.S.M. & Greitzer, S.L. (1967). Fagnano's Problem, pp. 88–89. Geometry Revisited. The Mathematical Association of America.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Hildebrandt, S.; & Tromba, A. (1985). Mathematics of Optimal Form. New York: Scientific American Library of W. H. Freeman & Co.
Rane, U. (2017). Fagnano's Problem: A Geometric Solution. At Right Angles, Vol. 6, No. 1 (March), pp. 21-24.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Fagnano's problem (Wikipedia)
Fagnano's problem (Cut The Knot)
Fagnano's problem by Unfolding (YouTube Video by Ujjwal Suryakant Rane)
Fagnano's Problem Path: Physical Solution (YouTube Video by Ujjwal Suryakant Rane)
Fagnano's Problem Path: Analytical Solution (YouTube Video by Ujjwal Suryakant Rane)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 12 Jan 2026.