The vesica has a long and genuinely documented life in Christian art and building, which is unusual among sacred-geometry claims and worth separating carefully from the claims that are not.
The mandorla is the almond-shaped frame of light drawn around an entire figure, as distinct from the nimbus around the head alone. It appears in Christian art from late antiquity and is standard by the Romanesque and Gothic periods, most often around Christ in Majesty above a west door, around Christ at the Transfiguration or the Ascension, and around the Virgin. The carved tympana at Moissac, Vezelay and Chartres are among the best known examples. The convention marks the figure as belonging to a different order of reality from the surrounding scene, rather than merely as being important.
Whether the mandorla is derived from the two-circle construction, or simply happens to be an almond shape, is a fair question, and art historians do not all answer it the same way. What is not in dispute is that Gothic masons drew with compasses constantly, and that circle-on-circle intersection is the generative move of window tracery. The pointed arch itself is two arcs struck from centres on the springing line, which is the vesica construction applied to an opening. Trefoils, quatrefoils and the late Fischblase forms are all built from arcs of the same kind.
Some claims go further than the evidence supports: that particular cathedrals were laid out on a vesica grid, or that the proportion encodes a specific doctrine. Those are argued rather than established, since medieval building accounts record dimensions and disputes but rarely the reasoning behind a geometric scheme. The safe statement is also the strong one. The construction is demonstrably the basis of the drawing technique, and the shape is demonstrably a standard Christian iconographic frame.