Latest revision as of 00:45, 18 February 2021

Carnot's Theorem states that in a triangle $ABC$ , the signed sum of perpendicular distances from the circumcenter $O$ to the sides (i.e., signed lengths of the pedal lines from $O$ ) is:

$OO_A+OO_B+OO_C=R+r$

$[asy] pair a,b,c,O,i,d,f,g; a=(0,0); b=(4,0); c=(1,3); O=circumcenter(a,b,c); i=incenter(a,b,c); draw(a--b--c--cycle); draw(circumcircle(a,b,c)); draw(incircle(a,b,c)); dot(i); dot(O); label("$A$",a,W); label("$B$",b,E); label("$C$",c,N); label("$I$",i,N); label("$O$",O,N); d=foot(O,b,c); dot(d); draw(O--d); label("$O_A$",d,N); draw(rightanglemark(O,d,b)); f=foot(O,a,b); dot(f); draw(O--f); draw(rightanglemark(O,f,a)); label("$O_C$",f,S); g=foot(O,c,a); dot(g); draw(O--g); draw(rightanglemark(O,g,a)); label("$O_B$",g,W); [/asy]$

where r is the inradius and R is the circumradius. The sign of the distance is chosen to be negative iff the entire segment $OO_i$ lies outside the triangle. Explicitly,

$OO_A+OO_B+OO_C=\frac{abc(|\cos{A}|+|\cos{B}|+|\cos{C}|)}{4|\Delta|}$

where $\Delta$ is the area of triangle $\Delta ABC$ .

Weisstein, Eric W. "Carnot's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CarnotsTheorem.html

Carnot's Theorem

Carnot's Theorem states that in a triangle $ABC$ with $A_1\in BC$ , $B_1\in AC$ , and $C_1\in AB$ , perpendiculars to the sides $BC$ , $AC$ , and $AB$ at $A_1$ , $B_1$ , and $C_1$ are concurrent if and only if $A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2$ .

Proof

Only if: Assume that the given perpendiculars are concurrent at $M$ . Then, from the Pythagorean Theorem, $A_1B^2=BM^2-MA_1^2$ , $C_1A^2=AM^2-MC_1^2$ , $B_1C^2=CM^2-MB_1^2$ , $A_1C^2=MC^2-MA_1^2$ , $C_1B^2=MB^2-MC_1^2$ , and $B_1A^2=AM^2-MB_1^2$ . Substituting each and every one of these in and simplifying gives the desired result.

If: Consider the intersection of the perpendiculars from $A_1$ and $B_1$ . Call this intersection point $N$ , and let $C_2$ be the perpendicular from $N$ to $AB$ . From the other direction of the desired result, we have that $A_1B^2+C_2A^2+B_1C^2=A_1C^2+C_2B^2+B_1A^2$ . We also have that $A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2$ , which implies that $C_1A^2-C_1B^2=C_2A^2-C_2B^2$ . This is a difference of squares, which we can easily factor into $(C_1A-C_1B)(C_1A+C_1B)=(C_2A-C_2B)(C_2A+C_2B)$ . Note that $C_1A+C_1=C_2A+C_2B=AB$ , so we have that $C_1A-C_1B=C_2A-C_2B$ . This implies that $C_1=C_2$ , which gives the desired result.

Carnot Extended

Let $P,Q,R$ be points in the plane of triangle $ABC$ . Then the perpendiculars from $P,Q,R$ to $BC,CA,AB$ respectively are concurrent if and only if $PB^2-PC^2+QC^2-QA^2+RA^2-RB^2=0$

Proof

Let $X,Y,Z$ be the feet of perpendiculars from $P,Q,R$ to $BC,CA,AB$ respectively. Note that $PB^2-PC^2=XB^2-XC^2$ from the application of pythogorean theorem to triangles $PXB,PXC$ . Now with similar relations for $Y$ and $Z$ , Carnot's theorem finishes the job!

Problems

Olympiad

$\triangle ABC$ is a triangle. Take points $D, E, F$ on the perpendicular bisectors of $BC, CA, AB$ respectively. Show that the lines through $A, B, C$ perpendicular to $EF, FD, DE$ respectively are concurrent. (Source)

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Difference between revisions of "Carnot's Theorem"

Latest revision as of 00:45, 18 February 2021

Contents

Carnot's Theorem

Proof

Carnot Extended

Proof

Problems

Olympiad

See also

@@ Line 1: / Line 1: @@
-'''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math> with <math>A_1\in BC</math>, <math>B_1\in AC</math>, and <math>C_1\in AB</math>, [[perpendicular]]s to the sides <math>BC</math>, <math>AC</math>, and <math>AB</math> at <math>A_1</math>, <math>B_1</math>, and <math>C_1</math> are [[concurrent]] [[if and only if]] <math>A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2</math>.
+'''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math>, the signed sum of [[perpendicular]] distances from the [[circumcenter]] <math>O</math> to the sides (i.e., signed lengths of the pedal lines from <math>O</math>) is:
-==Proof==
+<math>OO_A+OO_B+OO_C=R+r</math>
-{{incomplete|proof}}
+<asy>
+pair a,b,c,O,i,d,f,g;
+a=(0,0);
+b=(4,0);
+c=(1,3);
+O=circumcenter(a,b,c);
+i=incenter(a,b,c);
+draw(a--b--c--cycle);
+draw(circumcircle(a,b,c));
+draw(incircle(a,b,c));
+dot(i);
+dot(O);
+label("$A$",a,W);
+label("$B$",b,E);
+label("$C$",c,N);
+label("$I$",i,N);
+label("$O$",O,N);
+d=foot(O,b,c);
+dot(d);
+draw(O--d);
+label("$O_A$",d,N);
+draw(rightanglemark(O,d,b));
+f=foot(O,a,b);
+dot(f);
+draw(O--f);
+draw(rightanglemark(O,f,a));
+label("$O_C$",f,S);
+g=foot(O,c,a);
+dot(g);
+draw(O--g);
+draw(rightanglemark(O,g,a));
+label("$O_B$",g,W);
+</asy>
+where r is the [[inradius]] and R is the [[circumradius]]. The sign of the distance is chosen to be negative iff the entire segment <math>OO_i</math> lies outside the triangle.
+Explicitly,
+<math>OO_A+OO_B+OO_C=\frac{abc(|\cos{A}|+|\cos{B}|+|\cos{C}|)}{4|\Delta|}</math>
+where <math>\Delta</math> is the area of triangle <math>\Delta ABC</math>.
+Weisstein, Eric W. "Carnot's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CarnotsTheorem.html
+=Carnot's Theorem=
+'''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math> with <math>A_1\in BC</math>, <math>B_1\in AC</math>, and <math>C_1\in AB</math>, [[perpendicular]]s to the sides <math>BC</math>, <math>AC</math>, and <math>AB</math> at <math>A_1</math>, <math>B_1</math>, and <math>C_1</math> are [[concurrent]] [[iff|if and only if]] <math>A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2</math>.
+====Proof====
+'''Only if:''' Assume that the given perpendiculars are concurrent at <math>M</math>. Then, from the Pythagorean Theorem, <math>A_1B^2=BM^2-MA_1^2</math>, <math>C_1A^2=AM^2-MC_1^2</math>, <math>B_1C^2=CM^2-MB_1^2</math>, <math>A_1C^2=MC^2-MA_1^2</math>, <math>C_1B^2=MB^2-MC_1^2</math>, and <math>B_1A^2=AM^2-MB_1^2</math>. Substituting each and every one of these in and simplifying gives the desired result.
+''' If:''' Consider the intersection of the perpendiculars from <math>A_1</math> and <math>B_1</math>. Call this intersection point <math>N</math>, and let <math>C_2</math> be the perpendicular from <math>N</math> to <math>AB</math>. From the other direction of the desired result, we have that <math>A_1B^2+C_2A^2+B_1C^2=A_1C^2+C_2B^2+B_1A^2</math>. We also have that <math>A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2</math>, which implies that <math>C_1A^2-C_1B^2=C_2A^2-C_2B^2</math>. This is a difference of squares, which we can easily factor into <math>(C_1A-C_1B)(C_1A+C_1B)=(C_2A-C_2B)(C_2A+C_2B)</math>. Note that <math>C_1A+C_1=C_2A+C_2B=AB</math>, so we have that <math>C_1A-C_1B=C_2A-C_2B</math>. This implies that <math>C_1=C_2</math>, which gives the desired result.
+=Carnot Extended=
+Let <math>P,Q,R</math> be points in the plane of triangle <math>ABC</math>. Then the perpendiculars from <math>P,Q,R</math> to <math>BC,CA,AB</math> respectively are concurrent if and only if <cmath>PB^2-PC^2+QC^2-QA^2+RA^2-RB^2=0</cmath>
+====Proof====
+Let <math>X,Y,Z</math> be the feet of perpendiculars from <math>P,Q,R</math> to <math>BC,CA,AB</math> respectively. Note that <math>PB^2-PC^2=XB^2-XC^2</math> from the application of pythogorean theorem to triangles <math>PXB,PXC</math>. Now with similar relations for <math>Y</math> and <math>Z</math>, Carnot's theorem finishes the job!
+=Problems=
-==Problems==
 ===Olympiad===
 <math>\triangle ABC</math> is a triangle. Take points <math>D, E, F</math> on the perpendicular bisectors of <math>BC, CA, AB</math> respectively. Show that the lines through <math>A, B, C</math> perpendicular to <math>EF, FD, DE</math> respectively are concurrent. ([[1997 USAMO Problems/Problem 2|Source]])
@@ Line 10: / Line 72: @@
 ==See also==
+[[Carnot's Polygon Theorem]]
+[[Japanese Theorem]]
 [[Category:Geometry]]
 [[Category:Theorems]]