In triangle ABC,with usual notation,observe t | vedclass

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(A) We know the standard properties of a triangle where $\Delta$ is the area,$s$ is the semi-perimeter,and $r, r_1, r_2, r_3$ are the inradius and exr

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Both $I$ and $II$ are true

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(A) We know the standard properties of a triangle where $\Delta$ is the area,$s$ is the semi-perimeter,and $r, r_1, r_2, r_3$ are the inradius and exradii respectively.For statement $(I)$: The product of the inradius and the three exradii is given by $r r_1 r_2 r_3 = \frac{\Delta}{s} \cdot \frac{\Delta}{s-a} \cdot \frac{\Delta}{s-b} \cdot \frac{\Delta}{s-c} = \frac{\Delta^4}{s(s-a)(s-b)(s-c)}$. Since $\Delta^2 = s(s-a)(s-b)(s-c)$,we have $r r_1 r_2 r_3 = \frac{\Delta^4}{\Delta^2} = \Delta^2$. Thus,$(I)$ is true.For statement $(II)$: The sum of the products of the exradii taken two at a time is $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$. This is a standard identity in triangle geometry. Thus,$(II)$ is true.Therefore,both statements are true.

In $\triangle ABC$,with usual notation,observe the two statements given below:
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

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In $\triangle ABC$,with usual notation,observe the two statements given below:$(I)$ $r r_1 r_2 r_3 = \Delta^2$$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$Which of the following is correct?