Two intersecting circles C₁ and C₂ have a common tangent which touches C₁ at S and C₂ at T. The two circles intersect at X and Y, where Y is nearer to ST than X is.

Determine (with proof) the ratio:
Area(Triangle XYS)/Area(Triangle XYT)

The line XY intersects the common tangent ST
at its midpoint M.
Let U and V be the perpendicular projections
of S and T onto line XY.
The right triangles SUM and TVM are congruent.
[XYS] = Area(triangle XYS) = |XY||SU|
[XYT] = Area(triangle XYT) = |XY||TV|
Therefore, [XYS]/[XYT] = 1
QED 

Posted by Bractals on 2015-08-01 11:17:35