CJam, 90 bytes
Try it online
CJam isn't really optimized for geometry, so the code isn't doing anything fancy. It uses the parametric representation for two of the lines passing through a corner and being orthogonal to the opposite side, and solves the 2x2 system of equations resulting from intersecting the two lines. Well, it only needs one of the resulting parameters, which is then used for calculating the intersection point.
Note that some of the signs in the intermediate values are wrong. Instead of flipping them with additional code, I kept them the way they are, and corrected for the wrong sign by changing additions to subtractions (or vice versa) when the value is used later. For example, if you see a
+ where you would normally expect a
- when calculating a determinant, that's because one of the terms has the wrong sign. In fact, both determinants have the wrong sign (at least relative to my paper notes), but that doesn't matter because they are divided by each other.
Since the input format was not precisely specified, I made it CJam friendly. The code expects 3 points in CJam array format, where the values have to be floating point. For the given example, the expected input is:
[0. 0.] [3. 6.] [6. 0.]
q~ Get and interpret input.
:C Store away the last point in variable C. Will be needed at the end
to calculate the intersection point.
2$] Repeat the first point, and wrap all points in an array.
_2ew Generate array with all pairs of points [[A B] [B C] [C A]].
::.- Calculate the vector difference of all point pairs [A-B B-C C-A].
These are the vectors for each side of the triangle.
:D;; Store the side vectors in a variable, and clear the stack.
Calculate determinant for first unknown in 2x2 system of equations.
Calculate main determinant of 2x2 system of equations.
/ Divide the two determinants. This gives the parameter of the
intersection point for the line starting at C and orthogonal to A-B.
_ Copy it because we'll need it twice.
Calculate x-coordinate of intersection point.
Calculate y-coordinate of intersection point.
S\ Put a space between the two values.