Thanks Gabby.
I found a solution using another library in combination with Cesium  https://github.com/chrisveness/geodesy
Here’s my solution for anyone who encounters the same problem:
let Cesium = require(“cesium”)
let geodesy = require(“geodesy”)
// Convert Cesium.Cartographic to geodesy.LatLonVectors class
function toLatLon(point) {
return new geodesy.LatLonVectors(
Cesium.Math.toDegrees(point.latitude), Cesium.Math.toDegrees(point.longitude));
}
// Create one geodesic
const start1 = new Cesium.Cartographic(80, 45);
const end1 = new Cesium.Cartographic(85, 45);
const geod1 = new Cesium.EllipsoidGeodesic(start1, end1);
// Create second geodesic
const start2 = new Cesium.Cartographic(83, 40);
const end2 = new Cesium.Cartographic(83, 49);
const geod2 = new Cesium.EllipsoidGeodesic(start2, end2);
/**

Get the bearing of the geodesic from the start point

Yes, you could probably use EllipsoidGeodesic.startHeading

I just prefer to use one library for all the calculations in case there is a difference in math
*/
const start1Vector = toLatLon(start1); // Create start1 geodesy.LatLonVectors
const end1Vector = toLatLon(end1) // Create end1 geodesy.LatLonVectors
const geod1Bearing = start1Vector.bearingTo(end1Vector); // Get bearing
// Do the same for the second geodesic
const start2Vector = toLatLon(start2);
const end2Vector = toLatLon(end2)
const geod2Bearing = start1Vector.bearingTo(end2Vector);
// Caluclate the intersection
const geoIntersect = geodesy.LatLonVectors.intersection(
start1Vector, geod1Bearing, start2Vector, geod2Bearing);
``
Try it with RunKit  https://runkit.com/hnipps/59e8c572ba85c70011243c55
Thanks,
Harry