Problem 12.1. Find the area of the triangle determined by the straight line with equation $x-y+1=0$ and the tangent lines to the graph of the parabola $y=x^{2}-4 x+5$ at its common points with the line.

Emil Kolev
12.1. The common points of the graphs of the line and the parabola are $A(1,2)$ and $B(4,5)$. The equations of the tangents to the graph of the parabola at $A$ and $B$ are $y=-2 x+11$ and $y=4 x-11$, respectively. The intersecting point of the two tangents is the point $C\left(\frac{5}{2},-1\right)$. The area of $\triangle A B C$ then equals $\frac{27}{4}$.