In a triangle ABC,the sides a, b, c are in A. | vedclass

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(C) Let the sides $a, b, c$ be in $A$.$P$.,so $2b = a + c$. We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}

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$H$.$P$.

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(C) Let the sides $a, b, c$ be in $A$.$P$.,so $2b = a + c$. We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$. To check if $r_1, r_2, r_3$ are in $H$.$P$.,we check if $\frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3}$ are in $A$.$P$. $\frac{1}{r_1} = \frac{s-a}{\Delta}$,$\frac{1}{r_2} = \frac{s-b}{\Delta}$,$\frac{1}{r_3} = \frac{s-c}{\Delta}$. Since $a, b, c$ are in $A$.$P$.,$s-a, s-b, s-c$ are also in $A$.$P$. because $s-a + s-c = 2s - (a+c) = 2s - 2b = 2(s-b)$. Thus,$\frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3}$ are in $A$.$P$.,which implies $r_1, r_2, r_3$ are in $H$.$P$.

In a $\triangle ABC$,the sides $a, b, c$ are in $A$.$P$. if and only if $r_1, r_2, r_3$ are in

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