Olympiad problems

AND:

OR:

NO:

Found problems: 2

KoMaL A Problems 2024/2025, A. 885

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\,\triangle ABC$ touches side $BC$. The circumcircle of $\triangle BDE$ meets line $AB$ again at point $K$, the circumcircle of $\triangle CDF$ meets line $AC$ again at point $L$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\,\triangle ABC$. [i]Proposed by Luu Dong, Vietnam[/i]

1999 BAMO, 5

Tags: angle bisector , Angle-chasing , geometry

Let $ABCD$ be a cyclic quadrilateral (a quadrilateral which can be inscribed in a circle). Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = \frac{C}{FD}$. Let $P$ be the point on the segment $EF$ such that $\frac{PE}{PF} = \frac{AB}{CD}$. Prove that the ratio between the areas of triangle $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.