Let AB and CD be two diameters of the circle. For an arbitrary point P on the circle, let R and S be the feet of the perpendiculars from P to AB and CD, respectively. Show that the length of RS is independent of the choice of P.
First, a little angle chasing shows the angle between the diameters is equal to angle RPS, so this is a fixed angle. Call this angle O.
Next extend PR to intersect the circle at a second point and form chord PX with midpoint R.
Ans extend PS to intersect the circle at a second point and form chord PY with midpoint S.
The length of chord XY depends only on angle O.
By the midpoint connector theorem, RS = XY/2.
Thus the length of RS depends only on angle O and is independent of P.
Note: Now that we've shown P doesn't matter, we can move P to A and show length RS = AO sin(O)
Posted by Jer
on 2021-10-11 11:06:30