Saturday, November 7, 2009

Fox 177

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


  1. 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)
    Now 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)