Proof The product of the segments of intersecting chords

0:00- [Instructor] All right, what we wanna do in this video is given this circle and these two intersecting chords in this circle, we wanna prove that the product of these subsegments of each of these chords are the same. Or another way to say it, that the length of segment AC times the length of segment CE is equal to the length of segment BC times the length of segment CD.

0:24If you're inspired, I encourage you to pause this video and play around with it if you like, but we're about to jump in. So the first thing I'm gonna do is set up two triangles and then try to prove that these triangles are similar. And then I'm gonna set up the ratios between all of the sides that we see there.

0:43And then with a little bit of algebra, we should hopefully get to this result. Now, I've just drawn these two new chords to construct these two triangles. So now let's set about showing that they are similar triangles. Well, the first thing I could show, we have vertical angles.

0:58We know that the measure of this angle is going to be the same as the measure of that angle. Let me write this down. Vertical angles. Vertical angles. We have seen that in many other videos. Maybe I'll write it this way. This tells us because these angles are vertical, the measure of angle ACD, ACD is equal to the measure of angle, I'm running out of space, equal to the measure of angle BCE.

1:27BCE. All right, I have one angle down. If I can get another set of corresponding angles that are congruent that have the same measure, then I'm set because if I have two equal angles, and frankly all of the angles are going to be equal, I'm definitely dealing with similar triangles.

1:40Well, there's several ways that we can tackle it, but one way is to see that angle, let me just set a new color, angle, ADB. Let me write it over here. Angle ADB and angle AEB, they both intersect the same arc. And if we have angles that are inscribed in a circle, and I think this is pretty cool, that intersect the same arc, then those angles are going to be congruent.

2:20So this is going to be congruent to angle, angle AEB, angle AEB, or I could have said that their measures are the same. And I could mark that here. Maybe I'll do that in another color. I will put some double arcs here. And so now we have two triangles that have two sets of corresponding angles that are congruent, that have the same measure.

2:44So we can say that by, let me write it here, so by angle angle, we now know that triangle ACD, ACD is similar to triangle BCE, triangle BCE. Well, if these two triangles are similar, then the ratio of corresponding sides is going to be the same. So another way that we could say that is the ratio of DC over CA, so the length of DC over the length of CA, that should be the same, so DC over CA should be the same as the length of, and I should have actually said CD just to keep things consistent, so CD over AC.

3:32I'm just gonna do this in alphabetical order. So CD over AC should be the same as CE, CE over BC. I'm just looking at the ratio of corresponding sides. And that's because, that's because these two triangles are similar. And from here, we can just do a little bit of algebra.

3:56One way to think about it is we can just multiply both sides by the length of AC and the length of BC. This is equivalent, some of you might have done cross multiplication before, this is why it works because we're multiplying both sides of this equation by the same values.

4:10So the equality holds. This AC cancels with this AC, this BC cancels with this BC, and then we are left with BC times CD, the length of BC times the length of CD, is equal to the length of CE times the length of AC, which is exactly the result we were looking for.

4:33If you just swap both of these sides of the equality, you get exactly, and then you can also obviously change your order of multiplication here, you get exactly the result that we wanted to prove.