Hello, MATNTRNG!

Here is Captain Black's excellent construction.

Draw a horizontal line

At construct angle , at construct angle

. . intersecting at

Code:

R
o
* *
* *
* *
* *
* *
* *
* *
* *
* α β *
P o * * * * * * * * * o Q

Construct the bisector of angle , intersecting at

On , measure off

Code:

R
o
* *
* * *
* *
* * *
* *
* o C *
* *
* * b *
* α β *
P o * * * * o * * * * o Q
S

Through , construct a line parallel to

. . intersecting at

Through , construct a line parallel to

. . intersecting at

Code:

R
o
* *
* *
* *
* *
* C *
* o *
* * * *
* * * *
* α * * β *
P o * * o * * * o * * o Q
A B

is the desired triangle.