[从头学数学] 第130节 全等三角形

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了筑基中期的修炼，
这次要修炼的目标是[全等三角形]。

正剧开始：

星历2016年03月10日 12:44:32, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[全等三角形]。

为了和[人叫板老师]一样来探究这个全等的条件，小伟下了大力气专门做了一个三角形的工具。

首先是需要用到的顶点排序：

<span style="font-size:18px;">this.angleDraw = function(array, style, scale) {
	style = style ? style : 'black';
	//array是一个存放二维坐标点序列的数组
	var a0 = [].concat(array);
	
	scale = scale ? scale : 1;
	var len = a0.length;
	
	if (scale != 1 && scale > 0) {
		for (var i = 0; i < len; i++) {
			for (var j = 0; j < 2; j++) {
				a0[i][j]*=scale;
			}
		}
	}
	
	//进行环状排序，这样传入的array就可以任意顺序放置坐标点。
	var a = this.angularSort(a0);	
	
	//分两次绘点和连线
	var tmp = [].concat(a);	
	this.pointDraw(tmp, style);
	tmp = [].concat(a);
	this.strokeDraw(tmp, style);
	
	
	var d1, d2, d3, angle;
	var x1,y1, x2, y2, x3, y3;
	var s;
	//坐标点编号
	var s0 = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ';
	//为每个点利用余弦定理求角
	for (var i = 0; i < len; i++) {
		if (i == 0) {
			x1 = a[len-1][0];
			y1 = a[len-1][1];
			x3 = a[i+1][0];
			y3 = a[i+1][1];
		}
		else if (i == len-1) {
			x1 = a[i-1][0];
			y1 = a[i-1][1];
			x3 = a[0][0];
			y3 = a[0][1];
		}
		else {
			x1 = a[i-1][0];
			y1 = a[i-1][1];
			x3 = a[i+1][0];
			y3 = a[i+1][1];
		}
		x2 = a[i][0];
		y2 = a[i][1];
		
		d1 = (x1-x2)*(x1-x2)+(y1-y2)*(y1-y2);
		d2 = (x2-x3)*(x2-x3)+(y2-y3)*(y2-y3);
		d3 = (x1-x3)*(x1-x3)+(y1-y3)*(y1-y3);
		
		angle = Math.acos((d1+d2-d3)/(2*Math.sqrt(d1*d2)))/Math.PI*180;

		s = angle.toFixed(2)+'°';

		//document.write(s+'<p>');
		//标注角度和顶点编号
		plot.fillText(s, x2, -y2-5, 100);
		plot.fillText(s0[i], x2, -y2+20, 20);
		
	}	
	

}</span>

当然，这个早就做好了，只是在这里贴出来备忘而已。

接下来是这个工具：

<span style="font-size:18px;">function Triangle() {
	this.edges = [];
	this.angles = [];
	
	//已知三条边
	this.know3edges = function(edges) {
		this.edges = edges;
		//角度为弧度单位
		//a边对应角
		this.angles.push(Math.acos((edges[1]*edges[1] + edges[2]*edges[2]-edges[0]*edges[0])/(2*edges[1]*edges[2])));
		
		//b边对应角
		this.angles.push(Math.acos((edges[0]*edges[0] + edges[2]*edges[2]-edges[1]*edges[1])/(2*edges[0]*edges[2])));
		
		//c边对应角
		this.angles.push(Math.acos((edges[0]*edges[0] + edges[1]*edges[1]-edges[2]*edges[2])/(2*edges[0]*edges[1])));	
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	
	}
	
	//已知两个角
	this.know2angles = function(angles, r) {
		this.angles = [angles[0]/180*Math.PI, angles[1]/180*Math.PI, 
					  (180-(angles[0]+angles[1]))/180*Math.PI];
		
		//设其中一边长度为10
		r = r > 0 ? r : 10;
		//A边
		this.edges.push(r);
		
		var angleA = this.angles[0];
			angleB = this.angles[1];
			angleC = this.angles[2];
			
		//B边
		this.edges.push(Math.sin(angleB)/Math.sin(angleA)*r);
		//C边
		this.edges.push(Math.sin(angleC)/Math.sin(angleA)*r);
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	}
	
	//已知2条边
	this.know2edges = function(edges, angle) {
		//如果没有指定两边的夹角，默认为90度
		angle = angle ? angle/180*Math.PI : Math.PI/2;
		
		var edgeC = Math.sqrt(edges[0]*edges[0]+edges[1]*edges[1]-2*edges[0]*edges[1]*Math.cos(angle));
		this.edges = [edges[0], edges[1], edgeC];
		
		var edgeA = this.edges[0],
			edgeB = this.edges[1];
		
		//角度为弧度单位
		//a边对应角
		this.angles.push(Math.acos((edgeB*edgeB + edgeC*edgeC-edgeA*edgeA)/(2*edgeB*edgeC)));
		
		//b边对应角
		this.angles.push(Math.acos((edgeA*edgeA + edgeC*edgeC-edgeB*edgeB)/(2*edgeA*edgeC)));
		
		//c边对应角
		this.angles.push(angle);	
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	
	}
		
	//返回角度和边信息的字符串
	this.info = function() {
		var angleLabel = ['C', 'A', 'B'];
		var edgeLabel = ['ab', 'bc', 'ac'];
		var s = '';
		
		for (var i = 0; i < 3; i++) {
			s += edgeLabel[i]+' : ';
			s += this.edges[i].toFixed(2)+' ; ';
		}
		
		for (var i = 0; i < 3; i++) {
			s += angleLabel[i]+' : ';
			s += (this.angles[i]*180/Math.PI).toFixed(2) + ' ; ';
		}
		
		return s;
	}

}</span>

只要已知两条边或是两个角，或是三条边，小伟总能画出个三角形来。

这样就可以跟着[人叫板老师]一起做了。先来三条边的：

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know3edges([2, 5, 6]);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', 2*r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

效果是：

但这个角度好像有微小差别，到底小伟有没有错呢？

先验证[人叫板老师]的角度：

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know2angles([51.34, 110.46], 5);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', 2*r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

再验证小伟的角度：

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know2angles([51.32, 110.49], 5);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', 2*r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

可以看到边长都是完全一样的，或许，这就是传说中的误差允许范围吧。

接着看：

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know2edges([4, 5], 75);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

这个完全一样。

当然，如果只给定两条边，小伟就可以默认它是直角三角形了：

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know2edges([4, 5], 75);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

这个倒很方便。

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  

	var triangle = new  Triangle();
	var a = triangle.know2angles([45, 75], 5);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', r);
	
	var s = triangle.info();
	plot.fillText(s, -250, -100, 500);
	
}
</span>

<span style="font-size:18px;">//xGlobal, yGlobal是传入自页面的全局地图坐标，不一定需要用上。
function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  
	
	var r0 = 10*r, r1 = r0*0.707, r2 = r1*1.414/1.848;
	
	shape.fillDraw(shape.nEdge(0, 0, r0, 4), 'blue');
	shape.fillDraw(shape.nEdge(0, 0, r1, 4, Math.PI/4), 'pink');
	
	var a = shape.paraquad(0, 0, 0, r2, r2, Math.PI/4);
	var tmp;
	
	for (var i = 0; i < 4; i++) {
		plot.save()
			.rotate(-Math.PI*3/8-Math.PI/2*i);
		tmp = [].concat(a);
		shape.fillDraw(tmp, 'red');
		plot.restore();
	}
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  
	
	var triangle = new  Triangle();
	var a = triangle.know2angles([22.5, 22.5], 5);
	var tmp = [].concat(a);
	shape.angleDraw(tmp, 'red', r);
	
	var s = triangle.info();
	plot.fillText(s, -250, 150, 500);
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 75;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  
	
	var a = [[-r, 0], [r, 0], [0, r], [0, -2*r]];
	
	shape.angleDraw(a, 'red');
	
}</span>

本节到此结束，欲知后事如何，请看下回分解。