Question 48
Let $ABC$ be a triangle with all angles greater than $45^{\circ}$ and less than $90^{\circ}$.
a. Prove that one can place inside $ABC$ three squares with the following properties:
- the three squares all have the same side;
- the three squares have a common vertex $K$ interior to the triangle;
- any two squares have no points in common other than $K$;
- each of the squares has two opposite vertices on the perimeter of $ABC$ and for the rest lies entirely inside the triangle $ABC$.
b. Let $P$ be the centre of the square with side $AB$ and exterior to $ABC$. Let $r_{C}$ be the reflection of the line through $C$ and $K$ in the bisector of $A\widehat{C}B$ from $C$. Prove that $P$ belongs to $r_{C}$.
a. Prove that one can place inside $ABC$ three squares with the following properties:
- the three squares all have the same side;
- the three squares have a common vertex $K$ interior to the triangle;
- any two squares have no points in common other than $K$;
- each of the squares has two opposite vertices on the perimeter of $ABC$ and for the rest lies entirely inside the triangle $ABC$.
b. Let $P$ be the centre of the square with side $AB$ and exterior to $ABC$. Let $r_{C}$ be the reflection of the line through $C$ and $K$ in the bisector of $A\widehat{C}B$ from $C$. Prove that $P$ belongs to $r_{C}$.