Ceva's theorem

TV(a,b,c) is the affine /simple ratio, i.e the signed value of |ac| / |ab|.

The underlying JavaScript code

JXG.Options.label.autoPosition = true;
JXG.Options.text.fontSize = 24;

var board = JXG.JSXGraph.initBoard("jxgbox", {boundingbox: [-5,5,5,-5], axis:false, showCopyright:false, showNavigation:false});

var a = board.create('point', [-4, 0], {name:'a'});
var b = board.create('point', [4, -3], {name:'b'});
var c = board.create('point', [1, 4], {name:'c'});

var s1 = board.create('segment', [a, b], {color: 'black'});
var s2 = board.create('segment', [a, c], {color: 'black'});
var s3 = board.create('segment', [c, b], {color: 'black'});

var as = board.create('glider', [3, 0, s3], {name:"a'"});
var bs = board.create('glider', [4, -3, s2], {name:"b'"});
var cs = board.create('glider', [1, 4, s1], {name:"c'"});

var s4 = board.create('segment', [a, as], {color: 'black'});
var s5 = board.create('segment', [b, bs], {color: 'black'});
var s6 = board.create('segment', [c, cs], {color: 'black'});

var TV = function(p,q,t) {
        var v = p.Dist(t) / p.Dist(q);
        if (JXG.Math.Geometry.isSameDir(p.coords, q.coords, p.coords, t.coords)) {
            return v;
        } else {
            return -v;
        }
    };

var txt = board.create('text', [-4.5, -4, function() {
        return "TV(a',c,b) * TV(b',a,c) * TV(c',b,a) = " + (TV(as,c,b)*TV(bs,a,c)*TV(cs,b,a)).toFixed(2); }]);