△ 三角形垂心计算器（含外心、重心、内心和欧拉线）Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line)
输入三角形三个顶点的坐标，求垂心（三条高的交点），并同时给出重心、外心、内心、三边长、三个角、面积和三角形类型（锐角、直角、钝角）。还会检验垂心、重心、外心是否在同一条欧拉线上。本地处理。Enter the coordinates of a triangle three vertices to get the orthocentre (where the altitudes meet), along with the centroid, circumcentre, incentre, side lengths, angles, area and type (acute, right or obtuse). It also checks that the orthocentre, centroid and circumcentre lie on the Euler line. Processed locally.
使用说明与常见问题How-to & FAQ
「三角形垂心计算器（含外心、重心、内心和欧拉线）」是什么？What is the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) tool?
输入三角形三个顶点的坐标，求垂心（三条高的交点），并同时给出重心、外心、内心、三边长、三个角、面积和三角形类型（锐角、直角、钝角）。还会检验垂心、重心、外心是否在同一条欧拉线上。本地处理。Enter the coordinates of a triangle three vertices to get the orthocentre (where the altitudes meet), along with the centroid, circumcentre, incentre, side lengths, angles, area and type (acute, right or obtuse). It also checks that the orthocentre, centroid and circumcentre lie on the Euler line. Processed locally.
如何使用「三角形垂心计算器（含外心、重心、内心和欧拉线）」？How do I use the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) tool?
在「三角形垂心计算器（含外心、重心、内心和欧拉线）」中，填写「A 的 x」、「A 的 y」、「B 的 x」、「B 的 y」、「C 的 x」等参数，结果会随输入实时计算并显示在下方。In the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line): Fill in "A x", "A y", "B x", "B y", "C x"; results are recalculated instantly as you type.
垂心是什么？What is the orthocentre?
从每个顶点向对边（或其延长线）作垂线，三条高线交于一点，这一点就是垂心。锐角三角形的垂心在内部，直角三角形在直角顶点，钝角三角形在外部。Drop a perpendicular from each vertex to the opposite side (or its extension); the three altitudes meet at the orthocentre. It is inside an acute triangle, at the right angle of a right triangle and outside an obtuse one.
怎么计算？How is it calculated?
本页先求外心 O（三边垂直平分线的交点），再用 H = A + B + C − 2O 得到垂心，这来自欧拉线关系 OH = 3 × OG。The page finds the circumcentre O (where the perpendicular bisectors meet) and then H = A + B + C − 2O, which follows from the Euler line relation OH = 3 × OG.
欧拉线是什么？What is the Euler line?
任意非等边三角形的垂心、重心、外心都在一条直线上，重心把垂心到外心的线段分成 2 : 1。等边三角形时三点重合。In any non-equilateral triangle the orthocentre, centroid and circumcentre lie on one line, with the centroid dividing orthocentre to circumcentre in the ratio 2 : 1. In an equilateral triangle the three points coincide.
能举个计算实例吗？Can you show a worked example?
例如在「三角形垂心计算器（含外心、重心、内心和欧拉线）」中填 A 的 x = 0，A 的 y = 0，B 的 x = 4，B 的 y = 0，C 的 x = 0，C 的 y = 3，结果区显示：垂心 H：(0, 0)；垂心 x：0；垂心 y：0；三角形类型：直角三角形（垂心在直角顶点）；重心 G：(1.333333, 1)；外心 O：(2, 1.5)，外接圆半径 2.5；内心 I：(1, 1)，内切圆半径 1；边长 a = BC, b = CA, c = AB：5, 3, 4；角 A, B, C：90°, 36.8699°, 53.1301°；面积：6；欧拉线：H、G、O 共线，且…For example, entering A x = 0, A y = 0, B x = 4, B y = 0, C x = 0, C y = 3 in the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) gives: Orthocentre H: (0, 0); Orthocenter x: 0; Orthocenter y: 0; Type: Right triangle (orthocentre at the right angle); Centroid G: (1.333333, 1); Circumcentre O: (2, 1.5), circumradius 2.5; Incentre I: (1, 1), inradius 1; Sides a = BC, b = CA, c = AB: 5, 3, 4…
哪些输入会被检查？Which inputs are validated?
「三角形垂心计算器（含外心、重心、内心和欧拉线）」会检查这些输入——「A 的 x」、「A 的 y」、「B 的 x」、「B 的 y」、「C 的 x」、「C 的 y」：必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) checks its inputs — "A x", "A y", "B x", "B y", "C x", "C y": required; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「三角形垂心计算器（含外心、重心、内心和欧拉线）」的设置：「A 的 x」默认 0；「A 的 y」默认 0；「B 的 x」默认 4；「B 的 y」默认 0；「C 的 x」默认 0；「C 的 y」默认 3。每次生成结果都按当前设置处理。Settings in the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) — "A x" default 0; "A y" default 0; "B x" default 4; "B y" default 0; "C x" default 0; "C y" default 3. The result always uses the current settings.
计算精度如何？How precise are the results?
「三角形垂心计算器（含外心、重心、内心和欧拉线）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「三角形垂心计算器（含外心、重心、内心和欧拉线）」免费吗？需要注册吗？Is the Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) free? Do I need an account?
「三角形垂心计算器（含外心、重心、内心和欧拉线）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Orthocenter Calculator (with Circumcentre, Centroid, Incentre and Euler Line) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.