Problem Let $DE$ and $DF$ be the perpendicular lines drawn from any point $D$ on the hypotenuse $BC$ of a right isosceles triangle $ABC$ to the sides $AB$ and $AC$, respectively, and let $M$ be the midpoint of BC. Then $△EMF$ is also an isosceles right triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem Of the two points of intersection between the two lines that trisect $∠B$ of an isosceles triangle $ABC$ and the perpendicular line $AD$ drawn from the vertex $A$ to the base $BC$, the one closer to $A$ is called $M$ and the other is called $N$. If the intersection point of the line $CN$…

Problem Let $E$ and $F$ be the points where the line that passes through a point $D$ on the base $BC$ of an isosceles triangle $ABC$ and is perpendicular to $BC$ intersects with $AB$ and $AC$ (or their extensions). Construct rectangles $EDBG$ and $FDCH$, and let $S$ and $T$ be the midpoints of $DG$ and…

Problem If perpendicular lines $PD$ and $PE$ are dropped from any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with $A$ as its vertex to the sides $AB$ and $AC$, then $AD+AE$ is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The difference between the perpendicular lines $PQ$ and $PR$ drawn from any point $P$ on the extension of the base $BC$ of an isosceles triangle $ABC$ to the sides $AB$ and $AC$ (or their extensions) is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The sum of perpendicular lines $PQ$ and $PR$ drawn from any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with $A$ as its vertex to $AB$ and $AC$ is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $Q$ and $R$ be the points where the perpendicular line passing through any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with vertex $A$ intersects with the sides $AB$ and $AC$ (or their extensions), respectively. Then, if $PQ+PR$ is always equal to the side $BC$, $$∠A=∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$…

Problem If the sum of perpendicular lines $PD$ and $PF$ drawn from any point $P$ on one side of an isosceles triangle $ABC$ with vertex $A$ to the other two sides is equal to the height $AH$, then $△ABC$ is an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Extend the side $AB$ of an isosceles triangle $ABC$ with $A$ as the vertex, take a point $D$ such that $AB=BD$, and let $E$ be the midpoint of $AB$, then $$CD=2CE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If two triangles $ABC$ and $DBC$ have a common base $BC$, and $AD$ is parallel to $BC$, and $△ABC$ is an isosceles triangle, then the perimeter of $△ABC$ is less than that of $△DBC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…