Even though material interests may drive the human behavior,
there are still worldly images and unworldly lyrics inspring novelty,
reminding them to do good.
How can you draw righteousness on a board?
Does it really matter if charity is concave or convex?
--- Fox Weber 
Let OS extended meet the circle in A. Now ang. ROT =(π/2-α)/2 = π/4-α/2. Hence, ang. ROS = π/4-α/2 + α = π/4 + α/2 while ang. TSA = ang. ORT since OSRT is concyclic. Ang. ORT = π/2 - ang. ROT = π/2 -(π/4-α/2)= π/4 + α/2 Hence ang. TSA = ang. ROS or OR is parallel to ST ......(1)
ReplyDeleteNow PR=RT (Equal tangents) and PR=OS (OPRS is a rect.) Hence, OS = RT. In triangles SOT & SRT, we've: OS = RT, OT = RS = Circle radius and ST is common. The two triangles are congruent and hence α = Φ ..............(2)
In Tr. STO, ST/OT = sin(α)/sin(TSA) and in triangle ORT we've, OR/OT = sin(π/2)/sinORT)=1/sin(TSA). Combining these eqns, we get ST/OR =sin(α). Now A(QST)/A(QOR)= (ST/OR)^2 since the triangles are similar. In other words,A(QST)/A(QOR)= [sin(α)]^2 = [sin(Φ)]^2 ......(3)
Ajit
Good work!
ReplyDelete