Creating triangles from points with given bearing and leg length using ArcGIS Desktop

ArcGIS may have its own operation to calculate a point on a specific bearing and distance, but a great source for these types of calculations is the Aviation Formulary at: https://edwilliams.org/avform.htm

In particular you want the "Lat/Long given radial and distance" :

Lat/lon given radial and distance

A point {lat,lon} is a distance d out on the tc radial from point 1 if:

lat=asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
IF (cos(lat)=0)
   lon=lon1      // endpoint a pole
ELSE
   lon=mod(lon1-asin(sin(tc)*sin(d)/cos(lat))+pi,2*pi)-pi
ENDIF

This algorithm is limited to distances such that dlon around less than one quarter of the circumference of the earth in longitude. A >completely general, but more complicated algorithm is necessary if greater distances >are allowed:

lat =asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
dlon=atan2(sin(tc)*sin(d)*cos(lat1),cos(d)-sin(lat1)*sin(lat))
lon=mod( lon1-dlon +pi,2*pi )-pi

Note that trigonometry functions are in radians - you'll have to convert to/from degrees (Radians = Degrees*PI/180 )

ArcGIS has a number of geodetic operators available in the Projection Engine exposed through ArcPy as methods on the PointGeometry type.

The so-called "Forward Problem of Geodesy" is solved by the PointGeometry.pointFromAngleAndDistance() request, with required parameters angle (in degrees) and distance (in meters), and optional parameter method (one of GEODESIC, GREAT_ELLIPTIC, LOXODROME, and PRESERVE_SHAPE -- PLANAR is not available for geographic coordinate systems).

The "Inverse" (aka "Reverse") problem is solved by PointGeometry.angleAndDistanceTo() with required parameter other (the target PointGeometry), and optional parameter method (same as above), and returns a tuple of angle (in degrees) and distance (in meters) from the reference point.

Note that ArcPy also has a Point object, but this is a different object, which just stores multidimensional vertices. A Point is used with a SpatialReference to construct a PointGeometry.

The full solution for computing distance from New York (JFK) to Tokyo (Narita) in WGS84, and calculating the halfway point between them would look like:

import arcpy
gcs = arcpy.SpatialReference(4326)
gcs.setFalseOriginAndUnits(-400.0,-400.0,10000000)

jfk  = arcpy.PointGeometry(arcpy.Point(-73.7781,40.6413),gcs)
nrt  = arcpy.PointGeometry(arcpy.Point(140.3929,35.7720),gcs)

bearing,distance = jfk.angleAndDistanceTo(nrt,'GEODESIC')
print(" Bearing = {:.2f} deg".format(bearing))
print("Distance = {:.2f} meters".format(distance))

midpoint = jfk.pointFromAngleAndDistance(bearing,distance*0.5,'GEODESIC')
print("Midpoint = {:4f},{:.4f}".format(
        midpoint.firstPoint.X,
        midpoint.firstPoint.Y))

Which results in:

 Bearing = -27.32 deg
Distance = 10853320.27 meters
Midpoint = -152.890242,69.4824

and feels right (10853 kilometer NNW polar route over Alaska).