let , $k_{1},k_{2}$ touches $k$ at points $X,Y$ respectively. and $l\cap AB=Z$ so, $AX,BY,CZ$ are concurrent at the orthocentre of triangle $ABC$ by ceva's theorem we have $\frac{AZ}{ZB}=\frac{r_{1}}{r_{2}}$ now let, $AO_{1}\cap BC=R$ ; $BO_{2}\cap AC=S$ then , $\frac{CS}{SA}\cdot \frac{AZ}{ZB}\cdot...
well, i have proved the first part in differrent way . here it is , let $BX\cap AC={X}'$ and the incircle touches $BC$ at $F$ now , apply menelus's $ \frac{AE}{E{X}'}\frac{{X}'X}{XB}\frac{BD}{DA}=1 \Leftrightarrow \frac{{X}'X}{XB}=\frac{E{X}'}{BD}=\frac{E{X}'}{BF}$ as , $\frac{{X}'X}{XB}=\frac{CX}{C...
Let $ABC$ be an acute angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively . Let $X$ and $Y$ be the point of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ eith the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $...