Before revision:   First, we show when it is possible how to construct geodesic triangles which either have one, two, or three interior right angles . Then we report a construction of triples of points for which the dual Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of dual geodesic triangles with doubly right angles .
After revision:  First, we show when it is possible how to construct geodesic triangles which either have one, two, or three interior right angles , whenever it is possible. Second, we report a construction of triples of points for which the dual Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of dual geodesic triangles with doubly right angles .