In triangle ABC, match each expression with its value. List I List II (A) If r/ r a =1/4, find tan(A/2)[tan(B/2)+tan(C/2)]. (P) 2/3 (B) Find the minimum of r a r b r c / r 3 . (Q) 3/4 (C) Find

Question

In triangle ABC, match each expression with its value. List I List II (A) If r/ r a ​ =1/4, find tan(A/2)[tan(B/2)+tan(C/2)]. (P) 2/3 (B) Find the minimum of r a ​ r b ​ r c ​ / r 3 . (Q) 3/4 (C) Find the maximum of ( a 2 + b 2 + c 2 )/( 3 R 2 ). (R) 1 (D) P lies inside ABC; AP,BP,CP meet BC,CA,AB at D,E,F. Find PD/AD+PE/BE+PF/CF. (S) 3 (T) 27

TestHub · Accepted Answer

Use r/ r a ​ =tan(B/2)tan(C/2)=1/4 and the sum of pairwise half-angle tangent products 1 to obtain A=3/4. Since ∑ 1 /rᵢ=1/r, AM-GM gives B≥27 with equality in the equilateral case. Also (4/3) ∑ s i n 2 A ≤3, so C=3. The three ratios in D are barycentric weights, whose sum is 1.

Mathematics - Properties of Triangle Question with Solution | TestHub

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Doubts & Discussion

List I	List II
(A) If r/ $r_{a}$ =1/4, find tan(A/2)[tan(B/2)+tan(C/2)].	(P) 2/3
(B) Find the minimum of $r_{a}$ $r_{b}$ $r_{c}$ / $r^{3}$ .	(Q) 3/4
(C) Find the maximum of ( $a^{2}$ + $b^{2}$ + $c^{2}$ )/( $3 R^{2}$ ).	(R) 1
(D) P lies inside ABC; AP,BP,CP meet BC,CA,AB at D,E,F. Find PD/AD+PE/BE+PF/CF.	(S) 3
	(T) 27