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: Worksheet & Teacher Notes: Triangle Altitudes.
Prerequisites
It is recommended that the following Rethinking Proof activities have been completed before this one:
Water Supply II: Three Towns (Concurrency of perpendicular bisectors)
Areas Ratios
Concurrency Conjecture
For the alternative proof on the 2nd page of the dynamic sketch below, knowledge of the properties of cyclic quadrilaterals is required.
Triangle Altitudes
Notes
1) Follow the instructions and the guided proof in the accompanying Worksheet: Triangle Altitudes to produce an explanatory proof in terms of the concurrency of the perpendicular bisectors of a triangle.
2) For an alternative proof, click on the 'Link to Cyclic quadrilateral' button to navigate to a new sketch.
3) Click on the 'Hints' buttons in the new sketch, and use your knowledge of the properties of cyclic quadrilaterals to complete an alternative proof of the concurrency of the altitudes.
4) Use Ceva's theorem to prove the concurrency of the altitudes.
5) The concurrency of the altitudes also follow directly from the following two theorems:
Power Lines Special Case: Altitudes of a Triangle
Carnot's (or Bottema's) Perpendicularity Theorem.
Explore More
Some more properties related to the altitudes and the orthocentre, or variations of them, are explored in:
Light Ray in a Triangle (Fagnano's Minimal Path) (Rethinking Proof activity)
A generalization of Neuberg's Theorem and the Simson line
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
A variation of Miquel's theorem and its generalization
Minimum Area of Miquel Circle Centres Triangle
The Orthocentre Quadrilateral of a Quadrilateral
Euler-Nagel line analogy
Reference
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Concurrency Conjecture (Rethinking Proof activity)
Water Supply I: Four Towns (Rethinking Proof activity)
Water Supply II: Three Towns (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon (Rethinking Proof activity)
Miquel (Rethinking Proof activity)
Area Ratios (Rethinking Proof activity)
Kosnita's Theorem
Fermat-Torricelli Point Generalization (aka Jacobi's theorem) plus Further Generalizations
Dual to Kosnita (De Villiers points of a triangle)
Another concurrency related to the Fermat point of a triangle plus related results
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's theorem)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Bride's Chair Concurrency
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
Nine-point centre & Maltitudes of Cyclic Quadrilateral
A side trisection triangle concurrency
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency and Euler line locus result
Haag Hexagon - Extra Properties
Concurrency, collinearity and other properties of a particular hexagon
Power Lines of a Triangle
Power Lines Special Case: Altitudes of a Triangle
Carnot's Perpendicularity Theorem & Some Generalizations
Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D
Anghel's Hexagon Concurrency theorem
Some Circle Concurrency Theorems
Three Overlapping Circles (Haruki's Theorem)
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
Van Aubel's Theorem and some Generalizations (See concurrency in Similar Rectangles on sides)
The quasi-circumcentre and quasi-incentre of a quadrilateral
External Links
Altitude (triangle) (Wikipedia)
Orthocenter (Wikipedia)
Concurrent lines (Wikipedia)
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, 5 Oct 2025; updated 8 Oct 2025; 13 Jan 2026.