i am trying to solve following problem: suppose that legs AB=BC=30 in isosceles triangle,and center of inscribed circle divides altitude into 12:5 part,our aim is to find base,my problem is that i dont know what is equal radius or there is know any angle,so could not understand how to solve it,even i could not use Pythagorean theorem,because it would take a long calculations,please help me guys,i am trying to solve it right now,but no success yet
Let $O$ be the centre of the incircle. Join $O$ to the three vertices.
It is not absolutely clear what ratio $12:5$ means. Let $M$ be the middle of the base. Maybe (i) $BO:OM=12:5$, or maybe (ii) $BO:OM=5:12$. (It will turn out that (ii) can't happen.)
We examine the consequences of (i). Let the base be $2y$ (I don't like fractions). Let $BO=12t$ and let $OM=5t$.
We calculate the area of triangle $BAC$ in two different ways. The area is half of base times height, so it is $(1/2)(2y)(17t)$, that is, $17yt$.
The area is also the sum of the areas of $\triangle BOA$, $\triangle BOC$, and $\triangle OAC$. This sum is $(1/2)(30)(5t)+(1/2)(30)(5t)+(1/2)(2y)(5t)$, which is $150t +5yt$. We conclude that $$17yt=150t+5yt.$$ Thus $12y=150$, and therefore $2y=25$.
Next we examine possibility (ii). The analysis is similar. We find pretty quickly that $y$ is "too big" (the Triangle Inequality is violated).
$\Delta OBD \sim \Delta AB'B$