I have the following question with me:
"Let $B_1$ and $C_1$ be points on the sides $AC$ and $AB$ of a triangle $ABC$. Lines $BB_1$ and $CC_1$ intersect at point $D$. Prove that a circle can be inscribed inside quadrilateral $AB_1DC_1$ if and only if the incircles of $ABD$ and $ACD$ are tangent to each other."
I start like this:
I introduce notations:
$$AB = c, AC = b, BC = a, AC_1 = x, AB_1 = z, B_1D = y, C_1D = w, BD = q, CD = p$$
To prove the above statement I basically need to prove that the statements $$x + y = z + w$$ and $$c + p = b + q$$
are equivalent. Performing a few computations involving bases I get the following equations :
$$qz(p + w) = bw(q + y)$$ and $$yc(p + w) = px(q + y)$$
How do I proceed from here? Any help is appreciated. Alternative solutions are also welcome