Carnot's Theorem

Let points A', B', and C' be located on the sides BC, AC, and respectively AB of ABC. The perpendiculars to the sides of the triangle at points A', B', and C' are concurrent iff

AC'² - BC'² + BA'² - CA'² + CB'² - AB'² = 0.

Proof

First we'll prove the "only if" direction.  Assume the three perpendiculars meet in point P.  The lines AP, BP, CP, A'P, B'P, C'P split the triangle into six right triangles, three pairs of which share a hypotenuse, while three other pairs share a leg.  By the Pythagorean theorem,

AC'² + C'P² = AP²
-BC'² - C'P² = -BP²
BA'² + A'P² = BP²
-CA'² - A'P² = -CP²
CB'² + B'P² = CP²
-AB'² - B'P² = -AP²

Adding up the six equations completes the proof.

Now, to tackle the "if" direction, assume

AC'² - BC'² + BA'² - CA'² + CB'² - AB'² = 0.

and let P be the point of intersection of A'P and B'P.  Drop a perpendicular DP from P to AB.  By the "only if" part, already proven,

AD² - BD² + BA'² - CA'² + CB'² - AB'² = 0.

and then by subtracting the last two equations, one from the other, we get

AD² - BD² = AC'² - BC'² 
(AD -BD)(AD+BD) = (AC' -BC')(AC'+BC') 
(AD -BD)(AB) = (AC' -BC')(AB) 
AD -BD = AC' -BC' 
AD -AC' = BD-BC'

Now, if D is between C' and B (or D is at or beyond B) then AD > AC' and BD < BC', which would give the left and right sides of the last equation different signs.  Likewise, if D is between C' and A (or D is at or beyond A) then AD < AC' and BD > BC', which is similarly impossible, so D=C'.

Note: There is nothing about either direction of this proof that precludes points A', B', and/or C' from being beyond the vertices of the triangle.  That is, point A', for example, may be on the extension of side BC, rather than on the BC itself.  In fact, as cut-the-knot points out, points A', B', and C' need not be even on the sides of the triangle, but rather anywhere on a line perpendicular to the corresponding side of a triangle.  Even in this case, Carnot's Theorem still holds.

Corollary 1: Perpendicular bisectors of the sides of a triangle concur in a point (the circumcenter.)

Since the perpendiculars bisect AB, BC, and CA,

AC' = BC', BA' = CA', and CB' = AB',
so AC'² = BC'² , BA'² = CA'² , and  CB'² - AB'² , yielding immediately the result that
AC'² - BC'² + BA'² - CA'² + CB'² - AB'² = 0, so by Carnot's theorem, the three perpendiculars concur on a point.

Corollary 2: The altitudes of a triangle concur in a point.

Letting AA', BB', and CC' be the altitudes of ABC, we see that each altitude is a leg of two right triangles, so that

CC'² + AC'² = AC²,
-CC'² - BC'² = -BC²,
AA'² + BA'² = AB²,
-AA'² - CA'² = -AC²,
BB'² + CB'² = BC²,
-BB'² - AB'² = -AB²

Now, just as we did to prove Carnot's theorem, we simply add up the six equations to get

AC'² - BC'² + BA'² - CA'² + CB'² - AB'² = 0, so by Carnot's theorem, the three altitudes concur on a point.

. . . . . . .  A page on Menelaus' Theorem would be nice, too.

. . . . . . A page on Stewart's Theorem ( http://mathforum.org/library/drmath/view/54814.html  ) would be nice, too

Internet References

http://www.cut-the-knot.org/pythagoras/Carnot.shtml is a really easy-to-follow proof, from which I borrowed liberally here.

http://www.cut-the-knot.com/Generalization/ceva.html has a java-enabled diagram on which you can drag the vertices of the triangle, and watch the various similar triangles in action.

Related pages in this website:

Summary of geometrical theorems

Triangles -- a home page for anything about triangles.

Isogonal Conjugates -- the relationship between the orthocenter and circumcenter of a triangle

The webmaster and author of the Math Help site is Graeme McRae.
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