Consider a circle O with a diameter AB, shown here in green. Draw a second circle (red) with diameter AC, such that C is on AB. Draw an isosceles triangle with base CB and third vertex D on circle O. Draw a third circle (X), tangent to the first three figures. Prove that the line from C to the center of circle X is perpendicular to AB.

(In reply to First Step by Leming)

first, I redrew the whole idea where the red circle is half the diameter of the green circle. This expands the base of the triangle, and it does work out to be isosceles. SO, these calculations should work for any sized red circle/tringle combo.

Printed out picture and inserted new points and lines. Assuming CX to be perpindicular, the angle between CD and CX would have to be 30 degrees (60 triangles). Now, length of HX is half of CX (sin 30), which means CX is 2/3 AB. (Knowing this made it a LOT easier to draw the blue triangle on mine, and it did work out!)

more to come......

Posted by Jim on 2004-09-09 11:44:50