Q1.If radius of circumcircle of triangle is 10 and that of incircle is 5, then the square of the sum of radii of the escribed circles is?
Answer:2025
Q2.Please see attachment
Q1: Hint: r1+r2+r3-r=4R (this can be proved by using the results for exradii, inradius and circumradius, one can refer to Loney's Trigonometry - 1 for this)
Q2: \(tan(A+B+C+D)=\frac {S_1-S_3}{1-S_2+S_4}\)
Here, S1=tan A+tan B+tan C+tan D (Sum of tangents taken one at a time)
S2= Sum of tangents taken two at a time
S3= Sum of tangents taken three at a time
S4= Product of all the four tangents
For a quadrilateral, \(A+B+C+D=2\pi\)
\(\Rightarrow tan(A+B+C+D)=0\)
S1=S3
\(\frac {tan A+tan B+tan C+tan D}{cot A+cot B+cot C+ cotD}=\frac {S_1}{S_3/S_4}\)
=S4
PS: The TeX codes for typing \(\frac {tan A+tan B+tan C+tan D}{cot A+cot B+cot C+ cotD}\) are \($\frac {tan A+tan B+tan C+tan D}{cot A+cot B+cot C+ cotD}\)$
Thank you sir.