# A segment divides a square into two polygons of with unequal incircles …

$1.$ Point A and B lie on the sides of a square, segment AB divides the square into two polygons each of which has an inscribed circle. One of the circles has radius 6 cm while the other one is larger. What is the difference, in cm, between the side length of the square and twice the length of segment AB ?

What must a polygon meet in order for it to have a inscribed circle? I am having trouble in just finding the right figure to work with in this problem, I think I'll have no problem finding the answer after the figure, but complete solutions are welcome

• the only configuration I can come up look likes this. The answer do not depends on the slope of the line and is $12$. – achille hui Jul 23 '18 at 14:40

Refer to the diagrams below. We focus on the bottom right quarter of the square.

The square has a side length $$2CM=CM+CN$$.

Now, =\begin{align*}\\&\implies CM+CN-2AB\\&\implies CM+CN-(AP+BP)-AB\\&\implies CM+CN-(AN+BM)-AB\\&\implies(CM-BM)+(CN-AN)-AB\\&\implies BC+AC-AB\\&\implies BC+AC-(AQ+BQ)\\&\implies BC+AC-(AE+BD)\\&\implies CD+CE=12\end{align*} Note that we applied equal tangent segments repeatedly.

• +1 very clever solution. – achille hui Jul 24 '18 at 8:43

Let the incircles have centers $$R$$ and $$S$$ and respective radii $$r$$ and $$s$$ (with $$r < s$$). Note that $$s$$ is half the side of the original square. Let $$T$$ be the the original square's vertex that completes right triangle $$\triangle ABT$$, and define $$c:= |AB|$$.

Zooming in ...

The four pairs of congruent right triangles allow us to write (naming squares by their diagonals) \begin{align} |\square ST| - |\square RT| &= 2\,|\triangle ABS| + 2\,|\triangle ABR| \tag{1} \\[6pt] s^2 - r^2 &= 2\cdot \frac12 sc + 2\cdot\frac12 rs \tag{2} \\[6pt] (s+r)(s-r) &= c(s+r) \tag{3} \\[6pt] s-r &= c \tag{4} \end{align}

The relation in the original question is merely $$(4)$$, "doubled". $$\square$$