If you plan on utilizing the base Geometry class, it is essential to include all your own vertices in the scene. This can be accomplished using the Vertex class in the following manner:
var geometry = new THREE.Geometry();
var v1 = new THREE.Vector3(0,0,0); // Vector3 used for specifying position
var v2 = new THREE.Vector3(1,0,0);
var v3 = new THREE.Vector3(0,1,0); // When dealing with 2D, ensure all vertices lie in the same plane (z = 0)
// Add new geometry based on the specified positions
geometry.vertices.push(v1);
geometry.vertices.push(v2);
geometry.vertices.push(v3);
Next, we must form a face from our vertices by providing the indices of the vertices in the order they were added above.
[EDIT]: As mentioned by mrdoob below, the vertices should also be added in counter-clockwise order.
https://i.sstatic.net/yvcAE.png
This was the issue with my fiddle. You can check out the corrected demo here.
geometry.faces.push(new THREE.Face3(0, 2, 1));
Lastly, create a material and merge it with your geometry to produce a Mesh. Even for 2D shapes, generating a mesh is necessary for display purposes.
var redMat = new THREE.MeshBasicMaterial({color: 0xff0000});
var triangle = new THREE.Mesh(geometry, redMat);
scene.add(triangle) // Added at the origin