Intersecting Secants Property and Two Circles of Unequal Radii

[nothing123]

i have a couple questions which i didnt want to continue on posting in the other thread or else it would get extremely hard to follow.

1. recall that the Intersecting Secants Property states that if two secants AB and CD intersect at an external point P, then PA x PB = PC x PD. well, i need to modify this theorem so that one of the secants turns into a tangent and derive a new theorem. any ideas?

2. Two circles of unequal radii intersect in X and Y. AXB is any line drawn through X meeting the circumferences again in A and B. Prove that ∠AYB remains the same size regardless of the position of AXB.

i simply need setting up the diagram for this. the instructions are somewhat hard to follow.

thanx in advance.

 

[HallsofIvy]

i have a couple questions which i didnt want to continue on posting in the other thread or else it would get extremely hard to follow.

1. recall that the Intersecting Secants Property states that if two secants AB and CD intersect at an external point P, then PA x PB = PC x PD. well, i need to modify this theorem so that one of the secants turns into a tangent and derive a new theorem. any ideas?

If AB is a tangent then A=B so PA= PB: (PA)²= PC x PD.

2. Two circles of unequal radii intersect in X and Y. AXB is any line drawn through X meeting the circumferences again in A and B. Prove that ∠AYB remains the same size regardless of the position of AXB.

i simply need setting up the diagram for this. the instructions are somewhat hard to follow.

thanx in advance.

Draw two intersecting circles, mark the points of intersection X and Y. Draw any line through X, mark the other points where that line intersects the two circles A and B, Draw AY and BY. There are, of course, many different lines through X. You want to prove that the measure of angle AYB is the same for all such lines. It might be helpful that that is equivalent to saying the arcs AX and XB have the same total measure for all such lines.