Question 109
Let $ABC$ be a right triangle and $D$ be a point on the hypotenuse $AB$.
a) Prove that the expression
$$
\frac{AC^{2}}{AD+CD} + \frac{BC^{2}}{BD+CD}
$$
does not depend on $D$.
b) Let $DE$ ($E \in AC$) and $DF$ ($F \in BC$) be the bisectors of $\angle ADC$ and $\angle BDC$, respectively. Find the minimum value of the expression
$$
\frac{CF}{CA} + \frac{CE}{CB}.
$$
a) Prove that the expression
$$
\frac{AC^{2}}{AD+CD} + \frac{BC^{2}}{BD+CD}
$$
does not depend on $D$.
b) Let $DE$ ($E \in AC$) and $DF$ ($F \in BC$) be the bisectors of $\angle ADC$ and $\angle BDC$, respectively. Find the minimum value of the expression
$$
\frac{CF}{CA} + \frac{CE}{CB}.
$$