MATHEMATICS BONUS FILES for faculty and students http://www.onu.edu/~mcaragiu/bonus_files.html NOTES ON THE NINE POINT CIRCLE, THE SIMSON LINE AND RELATED TOPICS MIHAI CARAGIU April 7, 009 m-caragiu@onu.edu ABSTRACT. These notes are based on the proofs of the nine point circle and the Simson line theorems prepared by the author for the 006 and 007 Summer Honors Institute camps held at the Ohio Northern University, and on the author s 007 October 6 talk presented at the Ohio MAA Fall Meeting held at Wittenberg University. PRELIMINARIES Let,, C,. The reflection wof about the line through and is w= + It is particularly useful to consider the case in which, are points on the unit circle, that is, = =, or = and =. The reflection w of then becomes w= + As a corollary, if = =, the projection of onto the line through, is + w p = = + +
THEOREM. Let. THE ORTHOCENTER H h be the orthocenter of the triangle ABC, with a = b = c =. Then h= a+ b+ c. PROOF. Check that h a c b: + h a b+ c b c b+ c h a = = = = c b c b c b c b c b Similarly h b c a and h c b a. 3. THE FEET OF THE ALTITUDES Now that we discussed about orthocenters, it makes sense to find out the complex numbers corresponding to the feet of the altitudes of our triangle ( a= b= c= ) ABC remember,. Recall that ( + + ) represents the projection of onto the line through and where = =. Thus the foot of the altitude from A is bc ( a+ b+ c bca) = a+ b+ c. Similarly, the foot of the altitude a ac ab from B is a+ b+ c, and the foot of the altitude from C is a+ b+ c. b c
4. THE NINE POINT CIRCLE So far we know i The ORTHOCENTER h= a+ b+ c i The CENTROID g = a+ b+ c 3 a+ b+ c What about e: = other than being the midpoint of OH? a+ b b+ c c+ a The midpoints,, of the sides AB, BC, CA are at a a+ b+ c a+ b c distance from e=. Indeed, e = =, etc. bc ac ab The feet of the altitudes a+ b+ c, a+ b+ c, a b c a b + + c bc bc are also at a distance from e : e a+ b+ c = =, etc. a a We need three more points to complete a beautiful theorem!
The midpoints of the three segments from the orthocenter to the vertices, ( a+ h), ( b+ h) and ( c+ h) are at a distance from e. a+ b+ c a Indeed, e ( a+ h) = ( a+ ( a+ b+ c) ) = =, etc. THEOREM. (THE NINE POINT CIRCLE, OR THE EULER'S CIRCLE) Given any triangle ABC, the nine points listed below all lie on the same circle (Euler's Circle) centered at the midpoint between the circumcenter and the orthocenter of the triangle: i i i The midpoints of the three sides of the triangle, The feet of the three altitudes of the triangle, and The midpoints of the three segments from the orthocenter to the vertices. 5. INSCRIPTIBLE QUADRILATERALS THEOREM 3. Let,,, C represent the vertices of an inscriptible 3 4 4 4 quadrilateral. Let E, E, E, E be the Euler circles of the triangles, 4 3 4, and,respectively, with centers e, e, e, e.then E, E, E, E have a common point. 3 4 4 PROOF. We may assume, as usual, that = = = =. 4
+ + + + + + + + e =, e =, e =, e = 3 4 3 4 4 4 e + + + = 4 Define :.. e e = e e = e e3 = e e4 = e belongs to each all Euler circles E, E, E, E. 4 e, e, e, e lie on the same (blue) circle of radius centered at e. 4 This will be the "Euler Circle of the inscriptible quadrilateral " 4 6. SIMSON LINES Consider a triangle AAA with A, A A, = = =, 3 3 and let P be an arbitrary point in the plane of the triangle AAA.
Let P p, P p, P p be the projections of P onto the sides 3 3 AA, AA and AArespectively. Then 3 3 p = + + 3 3 p = + + 3 3 p3 = + + THEOREM 4. The three projections, p, p, p are collinear if and only if =, that is, if and only if P is on the circumcircle of the triangle AAA. PROOF p p = + + 3 3 + + 3 3 = 3 p3 p = + + + + 3 3 = 3 p, p, p collinear ( ) ( ) p p p p = p p p p 3 3 3 = 3 3 3 3 = 3 ( ) 3 ( 3)( ) = = = 3 3 0
7. AN EXTENSION OF SIMSON s THEOREM With the previous notation in place, we want to characterie the set of all PPP points P such that the (oriented) area of the triangle by the projections of P onto the sides of AA A is a given constant. determined Area ( ppp3) = Im ( p3 p)( p p) ( p3 p)( p p) ( p3 p)( p p) = 4i RECALL: p3 p = ( 3 )( ) and p p = ( )( 3 ) Thus, Area ( ppp ) = 3 ( 3 )( )( )( 3 ) ( 3 )( )( )( 3 ) 6i = ( 3 )( ) ( )( 3 ) 6i 3 3 ( p pp) We will eventually get to ( )( )( ) = 3 3 Area 6i 3 3 { C } C Let U = : =. DEFINE f : U, f,, = FACT. f,, is PURELY IMAGINARY for all,, U. PROOF. f,, ( )( )( ) = = (,, ) ( )( )( ) 3 3 ( )( 3)( 3 ) 3 3 = = f
= ( Δ ) = = ( Δ ) = ( Δ ) FACT. f,, 4 Area PROOF. f,, 4 R Area 4 Area = σ σ σ ( ) () () 3 ( σ ) FACT 3. f,, is antisymmetric: f,, f,, ( ) = = ( ) FACT 4. f, i, 4i 4i Area, i, = ( Δ ) CONCLUSION: f,, 4iArea inv ( ppp) ( )( )( ) ( )( )( ) = 3 3 Area 6i 3 ( ) 3 3 THEOREM 5. Area & = 4iArea Δ ( p pp) = Area ( Δ) 4 This represents an extension of Simson s Theorem! EXAMPLE: In the special case = 0, p, p, p are the midpoints of the sides of Δ and Area pp p = Area Δ 4.
8. EULER CIRCLES AND SIMSON LINES RECALL: inscriptible quadrilateral, E, E, E, E Euler circles 4 4 of the triangles,, and, e, e, e, etheir centers. 3 4 3 4 4 4 Then e= + + + is the center of the Euler circle or 4, 4 of radius, containing e, e, e, e. 4 CONNECTION WITH SIMSON LINES: If 4 is an inscriptible quadrilateral, the 4 Simson lines: of with respect to,of with respect to,of with 4 3 4 3 4 4 ect to respect to and of with resp the center e of the Euler circle of. 4, are concurrent, all passing through
AN EQUATION FOR THE SIMSON LINE Let be on the circumcircle of the triangle. The three projections, p, p, p of onto the lines,, are collinear, belonging to the Simson line l of with respect to. 3 3 The equation of is t p t p = = l t p p t p p pp pp p p p p p p pp pp t t = C, where C:= p p p p 3 RECALL: p p = ( )( 3 ) = ( ) ( )( 3 ) = p p = ( )( 3 ) = = ( )( 3 ) 3 3 p p Thus t t = C t t = C p p 3 Set t = p = 3 to determine C. + + 3 3 We get C = 3 3 + + + + = = ( + + + 3) ( + + + 3) Therefore the equation of the Simson line of with respect to Δ is + + + 3 + + + 3 + + + 3 t t = passes through t. = THEOREM 6 (ON SIMSON LINES AND EULER CIRCLES): If is an inscriptible quadrilateral, the 4 Simson lines: of with respect to, 4 4 34 3 4 and of 4with respect to 3, are of with respect to,of with respect to + + + e= 4 concurrent, all passing through the center of the Euler circle of 4. REFERENCES:. I.M. YAGLOM, Complex Numbers in Geometry, Academic Press, 968.. LIANG-SHIN HAN, Complex Numbers & Geometry, MAA, 994. 3. MIHAI CARAGIU, Geometry with Complex Numbers talk at Wittenberg University, September 007 http://www.onu.edu/~mcaragiu/oct6geometry.ppt