Problem In a right isosceles triangle $ABC$ with $∠B=∠R$, let $D$ and $E$ be the feet of the perpendicular lines drawn from a point $P$ on hypotenuse $AC$ to $AB$ and $AC$, respectively, and $F$ be the foot of the perpendicular line drawn from $P$ to $DE$. Then, the line $PF$ always passes through a…

Problem Construct squares $ABDE$ and $ACFG$ with the sides $AB$ and $AC$ of a right triangle $ABC$ with $∠B=∠R$, and let $H$ be the intersection point between the extension of $BA$ and $EG$. Then, $$BC=2AH.$$   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Take a point $D$ inside the isosceles right triangle $ABC$ with $∠C=∠R$, so that $AD=AC$ and $∠CAD=30°$. Then $$BD=CD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the point symmetrical to $A$ with respect to $DC$ is…

Problem In a rectangular triangle $ABC$ with $B$ as the right-angled vertex, if $D$ and $E$ are the feet of perpendicular lines drawn from $A$ to the bisectors of $∠C$ and its exterior angle, then $DE$ bisects $AB$ vertically. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In a rectangular triangle $ABC$, let $F$ be the intersection point of the perpendicular line $BE$ drawn from the right-angled vertex $B$ to $AC$ and the bisector $AD$ of $∠A$, and let $G$ be the intersection point of $BC$ with a line passing through $F$ and parallel to $AC$. Then,$$BF=BD=GC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$…

Problem In a rectangular triangle $ABC$ with $∠A$ as a right angle, if $D$ and $E$ are placed on the hypotenuse $BC$ such that $BD=AB$ and $CE=AC$, then $∠CAD$ and $∠BAE$ are equal to half of $∠B$ and $∠C$ respectively. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Draw a line parallel to the hypotenuse $BC$ from the vertex $A$ of the right isosceles triangle $ABC$, and place a point $D$ on it so that $BD=BC$. Let $E$ be the intersection point of $BD$ and $AC$. Then,$$CD=CE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Draw a straight line $AD$ parallel to the side $BC$ from the vertex $A$ of a right triangle $ABC$ with $∠C=∠R$, and let $F$ be the intersection point of the sides $AC$ and $BD$. Then, if $FD=2AB$, $$∠ABF=2∠FBC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $D$ be the foot of the perpendicular line drawn from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the hypotenuse $BC$. Then, the sum of the diameters of the inscribed circles of $△ABC, \ △ABD$ and $△ACD$ is equal to twice the length of $AD$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem In a right triangle $ABC$, $$∠A=∠R.$$Let $D$ be the foot of the perpendicular line drawn from $A$ to $BC$, $E$ be the point where the bisector of $∠B$ intersects $AC$, $F$ be the foot of the perpendicular line drawn from $E$ to $BC$, and $G$ be the point of intersection of $AD$ and $BE$.…