All over the map
Midpoints are another facet of symmetry in astrology.
They may or may not include conventional geometrical aspects. They are formed when one planet subtends two other planets or angles by identical arcs. The classic 'Finger of Fate' is an example: Planet A is simultaneously quincunx planets C and D which are in sextile (60 degrees) to each other. The three form an isosceles triangle. A is directly opposite the true midpoint of C/D.
A selection of The Queen's direct midpoints is illustrated here: these involve no conventional aspects, but her power and responsibility as Mother of the Nation is vividly shown in the Moon at the direct midpoint of Sun/Chiron-Saturn/Midheaven.
A 90-degree dial would pick out squares to midpoints in addition to those shown here. There is no theoretical limit to the fine-tuning that can be done; and a favourite of mine is the 15-degree dial.
No less than ten midpoints are stimulated in Elizabeth's chart by this lovely Venus: Sun-Moon (!), Chiron-Ceres, both Nodes-Neptune, both Nodes-Jupiter, Ascendant & Descendant-Neptune, Ascendant & Descendant-Mars; a very romantic, joyous constellation set on the physical and spiritual commitments of marriage. She absolutely adores him.
Midpoint synastry extends of course into our relationship with events, through the impact of transits to our own midpoints, and indeed the transits of current midpoints to our planets and angles.
Here is the impact of transiting Mars - assault, pain, blood - on the nativity of Jesus. The midpoints affected, involving his Sun/Jupiter and Nodes, Venus and the meridian - also Ceres - and Moon-Uranus, confirm his agony as the supreme act of love for which he came into incarnation.