23. Exemplar example problems IV

1. If $d_1$, $d_2$ $(d_2 > d_1)$ be the diameters of two concentric circles and $c$ be the length of a chord of a circle which is tangent to the other circle, prove that $d_2^2 = c^2 + d_1^2$.

 

2. If $a$, $b$, $c$ are the sides of a right triangle where $c$ is the hypotenuse, prove that the radius $r$ of the circle which touches the sides of the triangle is given by $r = \frac{a + b - c}{2}$.