Question

Describe the relationship between the segments made when secant lines intersect outside a circle. Use the following image as reference.

Answer 1

Let the extensions of secants AD and BC meet at point P.

Let the measure of ar DC be 2a, then m(DAC)=m(DBC)=a, since angles DAC and DBC are both inscribed angles intercepting arc DC.

m(P)=b.

Then triangles APC and BPD are similar, because they have 2 pairs of common angles.

from the similarity of APC and BPD, we can write the ratios:

$\frac{AP}{BP} = \frac{PC}{PD} = \frac{AC}{BD}$