find coordinates of fourth point on scalene triangle given 3d coordinates of corners

Question

You have a scalene triangle in 3d space, you know the coordinates of each of its points.

drawing a line perpendicular to the hypotenuse through the opposite corner, you split the scalene triangle into two right triangles.

How would one find the coordinates of the point at the intersection of the hypotenuse and this line, given the 3d coordinates of the other three points?

Answer 1

For triangle $ABC$ with known coordinates of all vertices, we can get all the side lengths $a,b,c$ and all the angles $\alpha,\beta,\gamma$.

There is almost no difference between 2D and 3D case. For example, the foot of the altitude from $C$ is found as

\begin{align} H_c&=\frac1{2c^2}\,((a^2+c^2-b^2)\cdot A+(b^2+c^2-a^2)\cdot B) ,\\ \text{or }\quad H_c&= \frac12\,(A+B)+\frac{a^2-b^2}{2c^2}\cdot(A-B) , \end{align}

since a convex linear combination of points in 3D works exactly the same way as in 2D.

Answer 2

Perhaps this question could have been better asked as "find coordinates of a point B on a line AC in 3d space" in the triangle ABC, I found the distance AD to point D via c*cos(A). The coordinates of D can then be described by:

x_D = x_A + (x_C - x_A) * (AD/AC)

y_D = y_A + (y_C - y_A) * (AD/AC)

etc.

Apologies that I could not understand the answers given.