Find the missing angle ABC triangle, knowing only angle BAC, side AB and the perimeter

Question

This question is similar to this post, but in my case the actual numbers do not matter, I'm interested in the equations.

I have an arbitrary triangle, of which only angle BAC and side AB are known. The sum of sides BC and AC (therefore the perimeter) is known and fixed.

I am looking for an equation that gives me the value of angle ABC.

Here's a screenshot, please remember that the exact dimensions are arbitrary, and are only there to show which parts are known. Triangle drawn in CAD

This is really driving me crazy, I would appreciate any help! Thank you!

Answer 1

This is a typical example of a problem which can be routinely solved using the $p,q$ method i.e., we assume that $A,B,C$ are $(0,0)$, $(c,0)$ and $(p,q)$. We can then find $p,q$ as functions of the three given quantities by solving $$ \frac qp =\tan A, c+((p-c)^2+q^2)^{1/2}+(p^2+q^2)^{1/2}=per.$$ This can be done by hand but I used Mathematica (the solution is not complicated but bad enough for it not to be feasible for me to type it in).

Once you know $p$ and $q$, you can easily find any triangle quantity, in particular $\angle ABC$.

Answer 2

With the great help of @user131781 and then mathway.com (to solve for x) I managed to express the x coordinate of the missing corner of the triangle as a function of triangle perimeter P and angle A, when |AB| is scaled to 1. The definition of y is trivial:

$$x=\frac{P\cdot (2−P)}{2\cdot (sec(A)−P\cdot sec(A)+1)}$$

$$y=x \cdot tan(A)$$

Reminder that the secant function is defined as$$sec(x)= \frac {1}{cos(x)}$$ (most computer languages/libraries don't have a built-in secant function).

The above formulation makes it trivial to calculate the third point of the triangle using any programming language. Once the third point is found, calculating the remaining sides and angles is trivial!

I plotted the 3 equations (2 from the correct answer, and the simplified x solution from mathway.com) in Desmos, and it's a thing of beauty! https://www.desmos.com/calculator/ghs0nxhhli

Note: The generic formulation with |AB| being variable (instead of normalized to 1) crashed every online solver I tried. Any help for a truly general solution would be appreciated!