( ABCde )[mathrm(AB)] perp[mathrm(AC)],[mathrm(A)+1] perp[mathrm(BC)] H B|-|H C|=|A H| |A C|=b |A B|=c (b^2+c^2)/(b^2)-c^(2)=?

To solve this problem, we need to understand the given information and the question being asked.<br /><br />Given:<br />- $\overline{AB}C$ is a triangle with vertices A, B, and C.<br />- $AC = 129911$<br />- $AH = I$<br />- $BC = HB$<br /><br />The question asks for the value of $\frac{b^2 + c^2}{b^2 - c^2}$, where $b$ and $c$ are the lengths of the sides of the triangle.<br /><br />To find the value of $\frac{b^2 + c^2}{b^2 - c^2}$, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.<br /><br />In this case, we can consider $\overline{AB}C$ as a right triangle with $AC$ as the hypotenuse and $AH$ and $I$ as the other two sides. Therefore, we have:<br /><br />$b^2 + c^2 = AC^2$<br /><br />$b^2 - c^2 = AH^2 - I^2$<br /><br />Substituting the given values, we have:<br /><br />$b^2 + c^2 = 129911^2$<br /><br />$b^2 - c^2 = AH^2 - I^2$<br /><br />Now, we can substitute these values into the expression $\frac{b^2 + c^2}{b^2 - c^2}$:<br /><br />$\frac{b^2 + c^2}{b^2 - c^2} = \frac{129911^2}{AH^2 - I^2}$<br /><br />Since we don't have the values of $AH$ and $I$, we cannot simplify this expression further. Therefore, the answer to the question is $\frac{129911^2}{AH^2 - I^2}$.