Morleys sætning.
Morleys sætning siger, at hvis vi i en vilkårlig trekant deler hver af de tre vinkler i tre lige store dele med “vinkeltredelingslinjer”, så vil den trekant, som disse i alt 6 linjer udstikker inde i den store trekant, være en ligesidet trekant. Du kan flytte trekantens hjørner A, B og C med musen. Det er pudsigt, at denne – egentlig ret simple – sætning ikke var kendt af de gamle grækere, men først er fundet i 1899 af amerikaneren Frank Morley (1860-1937).
Figure figure = Position [0,0] Size[x,y] Origin[x/3,y/2] Unit x/12 Color "white";
/*
Axes axes = Color "black";
Grid grid = Color "blue";
Units units = Color "black";
*/
Point A = [-2,0] Free Size 2.5 Color "red";
Point B = [3,-3] Free Size 2.5 Color "red";
Point C = [5,3] Free Size 2.5 Color "red";
Label laba = "A" At A Offset [10,-10] Color "black";
Label labb = "B" At B Offset [10,-10] Color "black";
Label labc = "C" At C Offset [10,-10] Color "black";
Line AB = Start A Dir B-A Size 1.5 Color "black";
Line BC = Start B Dir C-B Size 1.5 Color "black";
Line CA = Start C Dir A-C Size 1.5 Color "black";
Variable angleA_ = angle(C-A) - angle(B-A);
Variable angleA = if angleA_ < -pi then angleA_+2*pi else if angleA_ > pi then angleA_-2*pi else angleA_ endif endif;
Variable angleA1 = angle(B-A) + angleA*1/3;
Variable angleA2 = angle(B-A) + angleA*2/3;
Variable angleB_ = angle(A-B) - angle(C-B);
Variable angleB = if angleB_ < -pi then angleB_+2*pi else if angleB_ > pi then angleB_-2*pi else angleB_ endif endif;
Variable angleB1 = angle(C-B) + angleB*1/3;
Variable angleB2 = angle(C-B) + angleB*2/3;
Variable angleC_ = angle(B-C) - angle(A-C);
Variable angleC = if angleC_ < -pi then angleC_+2*pi else if angleC_ > pi then angleC_-2*pi else angleC_ endif endif;
Variable angleC1 = angle(A-C) + angleC*1/3;
Variable angleC2 = angle(A-C) + angleC*2/3;
Point A1 = intersect( B, B+[cos(angleB1),sin(angleB1)], C, C+[cos(angleC2),sin(angleC2)]) Size 2.5 Color "blue";
Point B1 = intersect( C, C+[cos(angleC1),sin(angleC1)], A, A+[cos(angleA2),sin(angleA2)]) Size 2.5 Color "blue";
Point C1 = intersect( A, A+[cos(angleA1),sin(angleA1)], B, B+[cos(angleB2),sin(angleB2)]) Size 2.5 Color "blue";
Line AB1 = Start A Dir B1-A Size 1 Color "black";
Line AC1 = Start A Dir C1-A Size 1 Color "black";
Line BA1 = Start B Dir A1-B Size 1 Color "black";
Line BC1 = Start B Dir C1-B Size 1 Color "black";
Line CA1 = Start C Dir A1-C Size 1 Color "black";
Line CB1 = Start C Dir B1-C Size 1 Color "black";
Line A1B1 = Start A1 Dir B1-A1 Size 1 Color "black";
Line B1C1 = Start B1 Dir C1-B1 Size 1 Color "black";
Line C1A1 = Start C1 Dir A1-C1 Size 1 Color "black";