[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10r = 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。