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?

  • $\begingroup$ It is the same way that you do for a triangle in 2D. $\endgroup$ – Math Lover Aug 1 at 14:20
  • $\begingroup$ That's not a helpful answer. I can find its distance from the three points, but i can't wrap my head around how to apply those numerical distances to 3d coordinates. What formula would i use? $\endgroup$ – Kama Aug 1 at 15:02

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.

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