Mannheim Circles
by liberator, Aug 23, 2014, 8:45 PM
In Rioplatense Olympiad 2013 Problem 6, we proved a result that concerned a concyclity relating to the orthocenter and the centroid:
A careful reader may have noticed that in the proof, there was in fact nothing special about the two centers aforementioned, other than the fact that the orthocenter may be described as the Miquel point of 
w.r.t 
, and that the centroid may be described as the intersection of the cevian rays defined by the circle intersections. This suggests a possible generalization of the question, and in fact, such a result exists. The following theorem was first proved by Mannheim in his Nouvelles Annales de Mathématiques (1890). The proof is very similar to that of Rioplatense Olympiad 2013 Problem 6, yet for completeness, it is included here.
Theorem (Mannheim) Let
be a triangle, and let 
be points on 
respectively. Let 
be points on 
, and denote 
. Then if 
, 
are concyclic.
![[asy]
/* First diagram for a generalization of Rioplatense Olympiad 2013 Problem 6, free script by liberator, 23 August 2014 */
unitsize(1.8cm);
defaultpen(fontsize(10pt));
pointpen=black;
pen dsp = rgb(0.4,0.6,0.8);
pen rcp = rgb(0.9,0,0);
pen gcp = rgb(0,0.6,0);
/* Initialize objects */
pair A = (-2.5,2.5);
pair B = (-3.5,-0.5);
pair C = (0.5,-0.5);
pair L = (-1.5,-0.5);
pair M = (-1,1);
pair N = (-2.8,1.6);
pair [] p = intersectionpoints(circumcircle(C,L,M), circumcircle(A,M,N));
pair P = p[1];
pair K = incenter(A,B,C);
pair [] x = intersectionpoints(A--K, circumcircle(A,M,N));
pair Ap = x[1];
pair Bp = IP(Line(B,K,-0.01, 2014), circumcircle(B,N,L));
pair Cp = intersectionpoint(C--K, circumcircle(C,L,M));
pair Q = IP(Line(P,Bp,-0.01,2014), circumcircle(A,M,N));
/* Draw objects */
draw(A--B--C--cycle, dsp+linewidth(1));
draw(A--K, dsp);
draw(B--Bp, dsp);
draw(A--Q, dsp);
draw(Bp--Q, dsp);
draw(circumcircle(A,M,N), rcp);
draw(circumcircle(B,N,L), rcp);
draw(circumcircle(C,L,M), rcp);
draw(circumcircle(Ap, Bp, Cp), gcp+dotted+linewidth(1));
/* Place dots on and label each point */
Drawing("A", A, dir(110));
Drawing("B", B, dir(200));
Drawing("C", C, dir(340));
Drawing("L", L, dir(-100));
Drawing("M", M, dir(-90));
Drawing("N", N, dir(130));
Drawing("A'", Ap, dir(60));
Drawing("B'", Bp, dir(0));
Drawing("C'", Cp, dir(110));
Drawing("K", K, dir(-90));
Drawing("P", P, dir(-45));
Drawing("Q", Q, dir(110));
[/asy]](//latex.artofproblemsolving.com/6/e/9/6e92a6ab4ada7e5313898e60367651d6ae2ea158.png)
Proof Define
the Miquel point of 
. Let 
. By Reim's theorem on 
, 
; by a converse of Reim's theorem, 
is cyclic: similarly, 
is cyclic, and the result follows. 
In fact, the reverse implication is also true. Mannheim did not actually prove this, but it is credited to him in R.A Johnson's Advanced Euclidean Geometry (1960).
Converse Let be a triangle, and let be points on respectively. Let be points on , and denote . Then if are concyclic, 
.
![[asy]
/* Second diagram for a generalization of Rioplatense Olympiad 2013 Problem 6, free script by liberator, 23 August 2014 */
unitsize(1.8cm);
defaultpen(fontsize(10pt));
pointpen=black;
pen dsp = rgb(0.4,0.6,0.8);
pen rcp = rgb(0.9,0,0);
pen gcp = rgb(0,0.6,0);
/* Initialize objects */
pair A = (-2.5,2.5);
pair B = (-3.5,-0.5);
pair C = (0.5,-0.5);
pair L = (-1.5,-0.5);
pair M = (-1,1);
pair N = (-2.8,1.6);
pair []p = intersectionpoints(circumcircle(C,L,M), circumcircle(A,M,N));
pair P = p[1];
pair K = incenter(A,B,C);
pair [] x = intersectionpoints(A--K, circumcircle(A,M,N));
pair Ap = x[1];
pair Bp = intersectionpoint(Line(B,K,-0.01, 2014), circumcircle(B,N,L));
pair Cp = intersectionpoint(C--K, circumcircle(C,L,M));
/* Draw objects */
draw(A--B--C--cycle, dsp+linewidth(1));
draw(A--K, dsp);
draw(B--Bp, dsp);
draw(C--K, dsp+dashed);
draw(circumcircle(A,M,N), rcp);
draw(circumcircle(B,N,L), rcp);
draw(circumcircle(C,L,M), rcp);
draw(circumcircle(Ap, Bp, Cp), gcp);
/* Place dots on and label each point */
Drawing("A", A, dir(110));
Drawing("B", B, dir(200));
Drawing("C", C, dir(340));
Drawing("L", L, dir(-100));
Drawing("M", M, dir(-90));
Drawing("N", N, dir(130));
Drawing("A'", Ap, dir(60));
Drawing("B'", Bp, dir(0));
Drawing("C'", Cp, dir(110));
Drawing("K", K, dir(-90));
Drawing("P", P, dir(-45));
[/asy]](//latex.artofproblemsolving.com/5/a/2/5a220137f8367636974f0d1ac3dc2c461f4f0143.png)
Proof Denote as before. By Miquel's theorem on 
w.r.t 
, it follows 
; by Miquel's theorem on 
, with 
on 
, it follows .
liberator wrote:
Let be an acute-angled scalene triangle, with centroid 
and orthocenter 
. The circle with diameter 
cuts the circumcircle of 
at 
, distinct from . Analogously define 
. Prove that 
are concyclic.
be an acute-angled scalene triangle, with centroid 
and orthocenter 
. The circle with diameter 
cuts the circumcircle of 
at 
, distinct from . Analogously define 
. Prove that 
are concyclic.
w.r.t 
, and that the centroid may be described as the intersection of the cevian rays defined by the circle intersections. This suggests a possible generalization of the question, and in fact, such a result exists. The following theorem was first proved by Mannheim in his Nouvelles Annales de Mathématiques (1890). The proof is very similar to that of Rioplatense Olympiad 2013 Problem 6, yet for completeness, it is included here.Theorem (Mannheim) Let
be a triangle, and let 
be points on 
respectively. Let 
be points on 
, and denote 
. Then if 
, 
are concyclic.![[asy]
/* First diagram for a generalization of Rioplatense Olympiad 2013 Problem 6, free script by liberator, 23 August 2014 */
unitsize(1.8cm);
defaultpen(fontsize(10pt));
pointpen=black;
pen dsp = rgb(0.4,0.6,0.8);
pen rcp = rgb(0.9,0,0);
pen gcp = rgb(0,0.6,0);
/* Initialize objects */
pair A = (-2.5,2.5);
pair B = (-3.5,-0.5);
pair C = (0.5,-0.5);
pair L = (-1.5,-0.5);
pair M = (-1,1);
pair N = (-2.8,1.6);
pair [] p = intersectionpoints(circumcircle(C,L,M), circumcircle(A,M,N));
pair P = p[1];
pair K = incenter(A,B,C);
pair [] x = intersectionpoints(A--K, circumcircle(A,M,N));
pair Ap = x[1];
pair Bp = IP(Line(B,K,-0.01, 2014), circumcircle(B,N,L));
pair Cp = intersectionpoint(C--K, circumcircle(C,L,M));
pair Q = IP(Line(P,Bp,-0.01,2014), circumcircle(A,M,N));
/* Draw objects */
draw(A--B--C--cycle, dsp+linewidth(1));
draw(A--K, dsp);
draw(B--Bp, dsp);
draw(A--Q, dsp);
draw(Bp--Q, dsp);
draw(circumcircle(A,M,N), rcp);
draw(circumcircle(B,N,L), rcp);
draw(circumcircle(C,L,M), rcp);
draw(circumcircle(Ap, Bp, Cp), gcp+dotted+linewidth(1));
/* Place dots on and label each point */
Drawing("A", A, dir(110));
Drawing("B", B, dir(200));
Drawing("C", C, dir(340));
Drawing("L", L, dir(-100));
Drawing("M", M, dir(-90));
Drawing("N", N, dir(130));
Drawing("A'", Ap, dir(60));
Drawing("B'", Bp, dir(0));
Drawing("C'", Cp, dir(110));
Drawing("K", K, dir(-90));
Drawing("P", P, dir(-45));
Drawing("Q", Q, dir(110));
[/asy]](http://latex.artofproblemsolving.com/6/e/9/6e92a6ab4ada7e5313898e60367651d6ae2ea158.png)
Proof Define
the Miquel point of 
. Let 
. By Reim's theorem on 
, 
; by a converse of Reim's theorem, 
is cyclic: similarly, 
is cyclic, and the result follows. 
In fact, the reverse implication is also true. Mannheim did not actually prove this, but it is credited to him in R.A Johnson's Advanced Euclidean Geometry (1960).
Converse Let
be a triangle, and let be points on respectively. Let be points on , and denote . Then if are concyclic, 
.![[asy]
/* Second diagram for a generalization of Rioplatense Olympiad 2013 Problem 6, free script by liberator, 23 August 2014 */
unitsize(1.8cm);
defaultpen(fontsize(10pt));
pointpen=black;
pen dsp = rgb(0.4,0.6,0.8);
pen rcp = rgb(0.9,0,0);
pen gcp = rgb(0,0.6,0);
/* Initialize objects */
pair A = (-2.5,2.5);
pair B = (-3.5,-0.5);
pair C = (0.5,-0.5);
pair L = (-1.5,-0.5);
pair M = (-1,1);
pair N = (-2.8,1.6);
pair []p = intersectionpoints(circumcircle(C,L,M), circumcircle(A,M,N));
pair P = p[1];
pair K = incenter(A,B,C);
pair [] x = intersectionpoints(A--K, circumcircle(A,M,N));
pair Ap = x[1];
pair Bp = intersectionpoint(Line(B,K,-0.01, 2014), circumcircle(B,N,L));
pair Cp = intersectionpoint(C--K, circumcircle(C,L,M));
/* Draw objects */
draw(A--B--C--cycle, dsp+linewidth(1));
draw(A--K, dsp);
draw(B--Bp, dsp);
draw(C--K, dsp+dashed);
draw(circumcircle(A,M,N), rcp);
draw(circumcircle(B,N,L), rcp);
draw(circumcircle(C,L,M), rcp);
draw(circumcircle(Ap, Bp, Cp), gcp);
/* Place dots on and label each point */
Drawing("A", A, dir(110));
Drawing("B", B, dir(200));
Drawing("C", C, dir(340));
Drawing("L", L, dir(-100));
Drawing("M", M, dir(-90));
Drawing("N", N, dir(130));
Drawing("A'", Ap, dir(60));
Drawing("B'", Bp, dir(0));
Drawing("C'", Cp, dir(110));
Drawing("K", K, dir(-90));
Drawing("P", P, dir(-45));
[/asy]](http://latex.artofproblemsolving.com/5/a/2/5a220137f8367636974f0d1ac3dc2c461f4f0143.png)
Proof Denote
as before. By Miquel's theorem on 
w.r.t 
, it follows 
; by Miquel's theorem on 
, with 
on 
, it follows . This post has been edited 3 times. Last edited by liberator, Jul 21, 2015, 11:18 AM
Reason: asy
Reason: asy
October 2018
April 2018