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 \(r = \frac{a + b - c}{2}\).