Carnot's Perpendicularity Theorem: S is any point inside (or outside) triangle DHL and perpendiculars are dropped from S to the sides, and squares are constructed as shown.
Investigate & Conjecture: What do you notice about the two sums of the areas of the green squares and of the pink squares? Formulate a conjecture.
Carnot's Perpendicularity Theorem & Some Generalizations
Historical Note: I'd previously known this result as Bottema's Theorem (1938) & had called it thus, but apparently it is originally due to the French mathematician, Lazare Carnot (1753-1823). This result also appears as a problem in Challenging Problems in Geometry by Alfred Posamentier & Charles Salkind (1996), pp. 14; 85-86. It is an exercise in C V Durell's New Geometry for Schools (1939), p. 287, Q26. Earlier still, it is in J W Russell's Sequel to Elementary Geometry (1907), p. 34, Sect 6 - a worked example. The converse result is an exercise in a French textbook, Traite de Geometrie by E Rouche and C de Comberousse (1900) vol 1, p. 395, Q254.
1) Can you explain why (prove) the theorem above is true? Hint: Connect S with the vertices and apply the theorem of Pythagoras to the six right triangles that are formed, group, and simplify.
2) Formulate the converse. Can you prove it?
Application & special cases
a) The concurrency of the perpendicular bisectors of a triangle, as well as of its altitudes, are special cases of the converse, which can be stated as follows: If DC2 + HG2 + LK2 = CH2 + GL2 + KD2, then the perpendiculars at C, G and K are concurrent at S.
b) The theorem of Pythagoras is a special case of Carnot's perpendicularity theorem. To visually see this in the figure above, drag triangle DHL until it is (approximately) a right triangle. Then drag the point S to place it at any one of the vertices other than the vertex of the (approximate) right angle.
c) The converse of the result can also be used to easily prove the concurrency of the Power Lines of a Triangle.
d) Can you generalize Carnot's Perpendicularity theorem further?
e) Click on the 'Similar Figure Generalization' button in the above sketch. Dynamic similar rectangles on the sides are shown, but just like the theorem of Pythagoras itself, the result generalizes to any similar figures, e.g. half circles or regular pentagons, etc. on the sides. Can you prove it in general? What about the converse? Is it true? Can you prove or refute the converse?
f) Click on the 'Polygon Generalization' button in the above sketch. A quadrilateral is shown but the result generalizes similarly to any polygon. Can you prove it in general? What about the converse? Is it true? Can you prove or refute the converse? What about similar figures on the sides of any polygon?
Some Additional Readings & Wikipedia Reference
Bottema, O. (1938). De Elementaire Meetkunde van het Platte Vlak. P. Noordhoff, Groningen- Batavia.
Cerin, Z. (2009). Rings of squares around orthologic triangles. University of Zagreb.
Jean-Pierre Ehrmann, JP. & van Lamoen, F. (2002). Some Similarities Associated with Pedals . Forum Geometricorum, Vol 2, 163–166. (Note that for some strange reason Firefox gives a warning for this URL, but it's safe & works fine on other browsers like Safari).
Wikipedia: Carnot's perpendicularity theorem.
Michael de Villiers, created 18 January 2009; modified/adapted to WebSketchpad, 3-4 September 2021.