Trigonometry – How to Find the Unknown Corner of a Triangle

trigonometry

I need to find an unkown corner of a triangle.
Image showing the situation.

There are 3 known points: $A$, $B$ & $C$.
Point $D$ is unknown.
$B$,$C$,$D$ forms the triangle.

The constraints are:

  • $D$ must lie on the extended line of $AB$.
  • The lines between $BD$ and $CD$ have the same length.

This can be solved iteratively as follows:

// Start off D at point B
D = B

// Loop until the distance between B and D is equal or larger than the distance between C and D.
while distance(B,D) < distance(C, D)
  D += normalize(B-A) * stepsize

// The loop is done, we know the answer with an error between 0 and stepsize
return D

This iterative process is potentially very slow when an exact answer is necessary. A single formula would be better

This problem can't be solved for all $AB$ vectors. It can only be solved if angle $∠ABC$ is greater than 90 degrees.

Best Answer

Hint: find midpoint of $BC$ and draw a line thru that point perpendicular to $BC$. The intersection of that line and extension of line $AB$ will give you point $D$. This is true because all points equidistant from $B$ and $C$ lie on the perpendicular bisector.

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