Is there any way of working out the coordinates at $D$ in the following picture? $AD$ is perpendicular to $BD$ in the plane $ABC$
2026-04-04 00:21:08.1775262068
Coordinates of a point in a three dimensional picture
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There is no unique $D$ point. The locus of such points is the circle in the $ABC$ plane with $AB$ as diameter. If you put the condition that $AD=BD$, then you are down to two points, one one the same side of $AB$ as $C$, one exactly opposite.
To construct such points, find the center of $AB$, and the radius (length of $AB$ divided by $2$. Then write the equation of the sphere with this radius and center, and the equation of the $ABC$ plane. Any point $D$ in the intersection verifies your first condition. To have $AD=BD$, you also add the equation of the plane perpendicular to $AB$ through the center, and solve the system of equations. You should get two solutions.