How can I create a line with three points with Python in QGIS?

You must first understand how PyQGIS handles geometry (Geometry Handling)

The most important element is the point:

QgsPoint(x,y)

and a line or a segment of line are composed of two points:

QgsGeometry.fromPolyline([QgsPoint(x1,y1),QgsPoint(x2,y2)]));

So to construct a line:

line_start = QgsPoint(50,50)
line_end = QgsPoint(100,150)
line = QgsGeometry.fromPolyline([line_start,line_end])

and with a memory layer (geometry only, without the attributes):

# create a new memory layer
v_layer = QgsVectorLayer("LineString", "line", "memory")
pr = v_layer.dataProvider()
# create a new feature
seg = QgsFeature()
# add the geometry to the feature, 
seg.setGeometry(QgsGeometry.fromPolyline([line_start, line_end]))
# ...it was here that you can add attributes, after having defined....
# add the geometry to the layer
pr.addFeatures( [ seg ] )
# update extent of the layer (not necessary)
v_layer.updateExtents()
# show the line  
QgsMapLayerRegistry.instance().addMapLayers([v_layer])

the result is:

with 3 points, just add it as a new feature:

newpoint = QgsPoint(143,125)
v_layer = QgsVectorLayer("LineString", "line_3pt", "memory")
pr = v_layer.dataProvider()
seg = QgsFeature()
seg.setGeometry(QgsGeometry.fromPolyline([line_start, line_end]))
# new feature: line from line_end to newpoint
seg = QgsFeature()
seg.setGeometry(QgsGeometry.fromPolyline([line_end, newpoint]))
pr.addFeatures( [ seg ] )
v_layer.updateExtents()
# add the line to 
QgsMapLayerRegistry.instance().addMapLayers([v_layer])

and the result is:

And with a for loop you can create a line with many segments: