Prove that internal angle bisectors of $\triangle ABC$ meet at a point.
The problem is that I have to prove this using the locus of a straight line and its properties, I can't use vectors.
The proof I can think of is very simple but extremely tedious.
Let the coordinate of triangle be $(0,0), (a,0), (x_0,y_0)$
Let side connecting $(0,0)\ \& \ (a, 0)$ be C, $(0,0) \ \& \ (x_0, y_0)$ be A and $(x_0,y_0) \ \& \ (a, 0)$ be B.
Then the equation to sides are,
$$C : y = 0 \ ; \ B : y(a – x_0) + xy_0 – ay_0 = 0\ ; \ A: xy_0 – yx_0 = 0$$
The general form of angle bisectors between two angles is
$${Ax + By + C\over \sqrt{A^2 + B^2}} = \pm {A_0x + B_0y + C_0\over \sqrt{A_0^2 + B_0^2}} $$
Using this equation, and some very tedious math I got,
$$ \operatorname{bisector(AC)} : y\left(\sqrt{x_0^2 + y_0^2} + x_0\right) -xy_0 = 0 \ ; \ \\ \operatorname{bisector(BC)} : y\left(\sqrt{(a -x-0)}- (a-x_0)\right)- xy_0+ay_0 = 0 \ ; \ \\ \operatorname{bisector(AC)} : x\left(\left(\sqrt{(a -x-0)}- (a-x_0)\right)y_0 – \sqrt{y_0^2 + x_0^2}y_0\right) – y\left(x_0\left(\sqrt{(a -x-0)}- (a-x_0)\right) + \sqrt{y_0^2 + x_0^2}(a-x_0)\right) + \sqrt{y_0^2 + x_0^2}ay_0 = 0$$
Now I just need to prove that these three lines are concurrent, for which I to prove that the determinant of the coefficient of these three lines equal $0$ .
I tried that but it does not come to zero, I probably lost somewhere in find the equations.
There has to be some other way of doing this, either by using some clever method of calculating the bisectors or by choosing the coordinates of the triangle such that we don't get such horrific equations.
Any help is appreciated.
Edit
Though the answers by @Joffran and @Mark is what I needed, I want to see if somebody can prove this by the method I described.
Best Answer
Consider the bisectors of two of the angles. Note that for any point on each bisector, the perpendicular distance to the two sides of the angle it bisects is equal, so at the intersection, the perpendicular distance to all three sides of the triangle is equal. Therefore this point is also on the third bisector line.
This intersection point is the center of the incircle - the common distance to each side is the radius of that circle.