Graphics, Figures & Tables ⇒ Morley’s triangle

[skysea575]

i try it:
In any triangle, trisector lines intersect in 3 points that are vertices of an equilateral triangle.

Intersections of trisector lines in any triangle are vertices of an equilateral triangle

another is:https://texample.net/tikz/examples/morleys-triangle/

Code: Select all

% 用tkz-euclide画莫勒定理
% Morley's triangle
% Author : 聂海波 hnkznhb@126.com
% Intersections of trisector lines in any triangle
% are vertices of an equilateral triangle

\documentclass{standalone}
\usepackage{tkz-euclide}
%--------------------------------------------------
%--------------------------------------------------
\begin{document}
% (莫勒定理) 在 $\triangle A B C$ 中, $P , Q , R$ 分别为三内角中两角的三分 线的交点, 如图所示, 则 $\triangle P Q R$ 为正三角形.


\begin{tikzpicture}%[scale=1.25,rotate=30]
	
	%----------------------------------------------------
	% Coordinates of A, B and C, the triangle vertices 
	%----------------------------------------------------
	\tkzDefPoints{0/0/A, 4/0/B, 2.8/2.8/C}%定义三角形3个顶点，初始点
	
	
	%----------------------------------------------------
	% FindAngle ACB,BAC,CBA
	%----------------------------------------------------
	\tkzFindAngle(A,C,B) \tkzGetAngle{angleACB}%定义三个内角，获得大小
	\tkzFindAngle(B,A,C) \tkzGetAngle{angleBAC}
	\tkzFindAngle(C,B,A) \tkzGetAngle{angleCBA}
	
	%----------------------------------------------------
	% Computing 1/3 of each angle,
	%Computing intersections of trisector lines 
	% Computing coordinates of vertices O, P and Q of 
	% the Morley's triangle 
	%----------------------------------------------------
	%以下定义旋转三等分点，利用两线交点确定P,Q,R
	\tkzDefPointBy[rotation=center B angle \angleCBA/3](C)
	\tkzGetPoint{CR}%旋转点
	\tkzDefPointBy[rotation=center C angle -\angleACB/3](B)
	\tkzGetPoint{BR}%旋转点
	\tkzInterLL(B,CR)(C,BR) \tkzGetPoint{P}%直线交点
	

	\tkzDefPointBy[rotation=center C angle \angleACB/3](A)
	\tkzGetPoint{AL}%旋转点
	\tkzDefPointBy[rotation=center A angle -\angleBAC/3](C)
	\tkzGetPoint{CL}%旋转点
	\tkzInterLL(A,CL)(C,AL) \tkzGetPoint{Q}%直线交点
	
	
	\tkzDefPointBy[rotation=center B angle -\angleCBA/3](A)
	\tkzGetPoint{AR}%旋转点
	\tkzDefPointBy[rotation=center A angle \angleBAC/3](B)
	\tkzGetPoint{BL}%旋转点
	\tkzInterLL(A,BL)(B,AR) \tkzGetPoint{R}%直线交点
	
	\tkzDrawSegments[color=blue,thin](B,P C,P)
	\tkzDrawSegments[color=cyan,thin](C,Q A,Q)
	\tkzDrawSegments[color=magenta,thin](A,R B,R)
	
	\tkzDrawPolygon[orange](P,Q,R)
	
	\tkzDrawPolygon[red](A,B,C)
	\tkzDrawPoints(A,B,C,P,Q,R)
	
	\tkzLabelPoints[below](A,B,R)
	\tkzLabelPoints[above](C)
	\tkzLabelPoints[left](Q)
	\tkzLabelPoints[right](P)
	
\end{tikzpicture}
\end{document}

[skysea575]

this is pic