[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.65,0:00:02.63,Default,,0000,0000,0000,,So I've got an\Narbitrary triangle here.
Dialogue: 0,0:00:02.63,0:00:04.83,Default,,0000,0000,0000,,We'll call it triangle ABC.
Dialogue: 0,0:00:04.83,0:00:07.13,Default,,0000,0000,0000,,And what I want to do\Nis look at the midpoints
Dialogue: 0,0:00:07.13,0:00:08.84,Default,,0000,0000,0000,,of each of the sides of ABC.
Dialogue: 0,0:00:08.84,0:00:12.05,Default,,0000,0000,0000,,So this is the midpoint of\None of the sides, of side BC.
Dialogue: 0,0:00:12.05,0:00:17.16,Default,,0000,0000,0000,,Let's call that point D. Let's\Ncall this midpoint E. And let's
Dialogue: 0,0:00:17.16,0:00:20.72,Default,,0000,0000,0000,,call this midpoint\Nright over here F.
Dialogue: 0,0:00:20.72,0:00:22.32,Default,,0000,0000,0000,,And since it's the\Nmidpoint, we know
Dialogue: 0,0:00:22.32,0:00:25.38,Default,,0000,0000,0000,,that the distance between BD\Nis equal to the distance from D
Dialogue: 0,0:00:25.38,0:00:28.21,Default,,0000,0000,0000,,to C. So this distance is\Nequal to this distance.
Dialogue: 0,0:00:28.21,0:00:31.38,Default,,0000,0000,0000,,We know that AE is equal\Nto EC, so this distance
Dialogue: 0,0:00:31.38,0:00:32.97,Default,,0000,0000,0000,,is equal to that distance.
Dialogue: 0,0:00:32.97,0:00:35.62,Default,,0000,0000,0000,,And we know that\NAF is equal to FB,
Dialogue: 0,0:00:35.62,0:00:39.15,Default,,0000,0000,0000,,so this distance is\Nequal to this distance.
Dialogue: 0,0:00:39.15,0:00:41.40,Default,,0000,0000,0000,,Instead of drawing medians\Ngoing from these midpoints
Dialogue: 0,0:00:41.40,0:00:42.94,Default,,0000,0000,0000,,to the vertices,\Nwhat I want to do is
Dialogue: 0,0:00:42.94,0:00:46.09,Default,,0000,0000,0000,,I want to connect these\Nmidpoints and see what happens.
Dialogue: 0,0:00:46.09,0:00:48.83,Default,,0000,0000,0000,,So if I connect them, I\Nclearly have three points.
Dialogue: 0,0:00:48.83,0:00:52.97,Default,,0000,0000,0000,,So if you connect three\Nnon-linear points like this,
Dialogue: 0,0:00:52.97,0:00:55.71,Default,,0000,0000,0000,,you will get another triangle.
Dialogue: 0,0:00:55.71,0:00:59.47,Default,,0000,0000,0000,,And this triangle that's formed\Nfrom the midpoints of the sides
Dialogue: 0,0:00:59.47,0:01:02.62,Default,,0000,0000,0000,,of this larger triangle-- we\Ncall this a medial triangle.
Dialogue: 0,0:01:06.80,0:01:08.88,Default,,0000,0000,0000,,And that's all nice\Nand cute by itself.
Dialogue: 0,0:01:08.88,0:01:10.06,Default,,0000,0000,0000,,But what we're going\Nto see in this video
Dialogue: 0,0:01:10.06,0:01:11.59,Default,,0000,0000,0000,,is that the medial\Ntriangle actually
Dialogue: 0,0:01:11.59,0:01:13.80,Default,,0000,0000,0000,,has some very neat properties.
Dialogue: 0,0:01:13.80,0:01:15.21,Default,,0000,0000,0000,,What we're actually\Ngoing to show
Dialogue: 0,0:01:15.21,0:01:19.29,Default,,0000,0000,0000,,is that it divides any triangle\Ninto four smaller triangles
Dialogue: 0,0:01:19.29,0:01:21.22,Default,,0000,0000,0000,,that are congruent\Nto each other,
Dialogue: 0,0:01:21.22,0:01:24.53,Default,,0000,0000,0000,,that all four of these triangles\Nare identical to each other.
Dialogue: 0,0:01:24.53,0:01:26.61,Default,,0000,0000,0000,,And they're all similar\Nto the larger triangle.
Dialogue: 0,0:01:26.61,0:01:28.43,Default,,0000,0000,0000,,And you could think\Nof them each as having
Dialogue: 0,0:01:28.43,0:01:31.32,Default,,0000,0000,0000,,1/4 of the area of\Nthe larger triangle.
Dialogue: 0,0:01:31.32,0:01:33.65,Default,,0000,0000,0000,,So let's go about proving it.
Dialogue: 0,0:01:33.65,0:01:35.70,Default,,0000,0000,0000,,So first, let's focus\Non this triangle down
Dialogue: 0,0:01:35.70,0:01:37.31,Default,,0000,0000,0000,,here, triangle CDE.
Dialogue: 0,0:01:37.31,0:01:40.20,Default,,0000,0000,0000,,And it looks similar\Nto the larger triangle,
Dialogue: 0,0:01:40.20,0:01:42.45,Default,,0000,0000,0000,,to triangle CBA.
Dialogue: 0,0:01:42.45,0:01:44.96,Default,,0000,0000,0000,,But let's prove it to ourselves.
Dialogue: 0,0:01:44.96,0:01:47.13,Default,,0000,0000,0000,,So one thing we can say is,\Nwell, look, both of them
Dialogue: 0,0:01:47.13,0:01:49.57,Default,,0000,0000,0000,,share this angle\Nright over here.
Dialogue: 0,0:01:49.57,0:01:53.65,Default,,0000,0000,0000,,Both the larger triangle,\Ntriangle CBA, has this angle.
Dialogue: 0,0:01:53.65,0:01:56.63,Default,,0000,0000,0000,,And the smaller triangle,\NCDE, has this angle.
Dialogue: 0,0:01:56.63,0:01:58.46,Default,,0000,0000,0000,,So they definitely\Nshare that angle.
Dialogue: 0,0:01:58.46,0:02:00.95,Default,,0000,0000,0000,,And then let's think about\Nthe ratios of the sides.
Dialogue: 0,0:02:00.95,0:02:14.24,Default,,0000,0000,0000,,We know that the ratio of CD\Nto CB is equal to 1 over 2.
Dialogue: 0,0:02:14.24,0:02:18.35,Default,,0000,0000,0000,,This is 1/2 of this entire\Nside, is equal to 1 over 2.
Dialogue: 0,0:02:18.35,0:02:22.98,Default,,0000,0000,0000,,And that's the same thing\Nas the ratio of CE to CA.
Dialogue: 0,0:02:22.98,0:02:27.04,Default,,0000,0000,0000,,CE is exactly 1/2 of CA,\Nbecause E is the midpoint.
Dialogue: 0,0:02:27.04,0:02:30.88,Default,,0000,0000,0000,,It's equal to CE over CA.
Dialogue: 0,0:02:30.88,0:02:33.30,Default,,0000,0000,0000,,So we have an angle,\Ncorresponding angles that
Dialogue: 0,0:02:33.30,0:02:35.15,Default,,0000,0000,0000,,are congruent, and\Nthen the ratios
Dialogue: 0,0:02:35.15,0:02:37.84,Default,,0000,0000,0000,,of two corresponding sides\Non either side of that angle
Dialogue: 0,0:02:37.84,0:02:38.48,Default,,0000,0000,0000,,are the same.
Dialogue: 0,0:02:38.48,0:02:45.01,Default,,0000,0000,0000,,CD over CB is 1/2, CE over CA\Nis 1/2, and the angle in between
Dialogue: 0,0:02:45.01,0:02:46.12,Default,,0000,0000,0000,,is congruent.
Dialogue: 0,0:02:46.12,0:03:03.46,Default,,0000,0000,0000,,So by SAS similarity, we\Nknow that triangle CDE
Dialogue: 0,0:03:03.46,0:03:08.07,Default,,0000,0000,0000,,is similar to triangle CBA.
Dialogue: 0,0:03:12.40,0:03:14.86,Default,,0000,0000,0000,,And just from that, you can\Nget some interesting results.
Dialogue: 0,0:03:14.86,0:03:16.32,Default,,0000,0000,0000,,Because then we\Nknow that the ratio
Dialogue: 0,0:03:16.32,0:03:20.23,Default,,0000,0000,0000,,of this side of the smaller\Ntriangle to the longer triangle
Dialogue: 0,0:03:20.23,0:03:21.88,Default,,0000,0000,0000,,is also going to be 1/2.
Dialogue: 0,0:03:21.88,0:03:23.88,Default,,0000,0000,0000,,Because the other two\Nsides have a ratio of 1/2,
Dialogue: 0,0:03:23.88,0:03:25.71,Default,,0000,0000,0000,,and we're dealing with\Nsimilar triangles.
Dialogue: 0,0:03:25.71,0:03:27.86,Default,,0000,0000,0000,,So this is going\Nto be 1/2 of that.
Dialogue: 0,0:03:27.86,0:03:32.75,Default,,0000,0000,0000,,And we know 1/2 of AB is just\Ngoing to be the length of FA.
Dialogue: 0,0:03:32.75,0:03:34.96,Default,,0000,0000,0000,,So we know that this\Nlength right over here
Dialogue: 0,0:03:34.96,0:03:37.91,Default,,0000,0000,0000,,is going to be the\Nsame as FA or FB.
Dialogue: 0,0:03:37.91,0:03:39.91,Default,,0000,0000,0000,,And we get that straight\Nfrom similar triangles.
Dialogue: 0,0:03:39.91,0:03:49.29,Default,,0000,0000,0000,,Because these are similar,\Nwe know that DE over BA
Dialogue: 0,0:03:49.29,0:03:51.73,Default,,0000,0000,0000,,has got to be equal\Nto these ratios,
Dialogue: 0,0:03:51.73,0:03:54.65,Default,,0000,0000,0000,,the other corresponding\Nsides, which is equal to 1/2.
Dialogue: 0,0:03:54.65,0:03:57.58,Default,,0000,0000,0000,,And so that's how we got\Nthat right over there.
Dialogue: 0,0:03:57.58,0:04:01.40,Default,,0000,0000,0000,,Now let's think about\Nthis triangle up here.
Dialogue: 0,0:04:04.07,0:04:05.19,Default,,0000,0000,0000,,We could call it BDF.
Dialogue: 0,0:04:07.80,0:04:09.61,Default,,0000,0000,0000,,So first of all, if\Nwe compare triangle
Dialogue: 0,0:04:09.61,0:04:12.93,Default,,0000,0000,0000,,BDF to the larger\Ntriangle, they both share
Dialogue: 0,0:04:12.93,0:04:15.78,Default,,0000,0000,0000,,this angle right\Nover here, angle ABC.
Dialogue: 0,0:04:15.78,0:04:17.81,Default,,0000,0000,0000,,They both have that\Nangle in common.
Dialogue: 0,0:04:17.81,0:04:20.28,Default,,0000,0000,0000,,And we're going to have\Nthe exact same argument.
Dialogue: 0,0:04:20.28,0:04:21.71,Default,,0000,0000,0000,,You can just look\Nat this diagram.
Dialogue: 0,0:04:21.71,0:04:27.12,Default,,0000,0000,0000,,And you know that the ratio\Nof BA-- let me do it this way.
Dialogue: 0,0:04:27.12,0:04:33.95,Default,,0000,0000,0000,,The ratio of BF to\NBA is equal to 1/2,
Dialogue: 0,0:04:33.95,0:04:36.72,Default,,0000,0000,0000,,which is also the\Nratio of BD to BC.
Dialogue: 0,0:04:40.82,0:04:42.83,Default,,0000,0000,0000,,The ratio of this\Nto that is the same
Dialogue: 0,0:04:42.83,0:04:45.71,Default,,0000,0000,0000,,as the ratio of this\Nto that, which is 1/2.
Dialogue: 0,0:04:45.71,0:04:47.36,Default,,0000,0000,0000,,Because BD is 1/2 of\Nthis whole length.
Dialogue: 0,0:04:47.36,0:04:49.50,Default,,0000,0000,0000,,BF is 1/2 of that whole length.
Dialogue: 0,0:04:49.50,0:04:51.86,Default,,0000,0000,0000,,And so you have\Ncorresponding sides
Dialogue: 0,0:04:51.86,0:04:53.86,Default,,0000,0000,0000,,have the same ratio\Non the two triangles,
Dialogue: 0,0:04:53.86,0:04:55.45,Default,,0000,0000,0000,,and they share an\Nangle in between.
Dialogue: 0,0:04:55.45,0:05:03.01,Default,,0000,0000,0000,,So once again, by\NSAS similarity,
Dialogue: 0,0:05:03.01,0:05:07.98,Default,,0000,0000,0000,,we know that triangle--\NI'll write it
Dialogue: 0,0:05:07.98,0:05:18.75,Default,,0000,0000,0000,,this way-- DBF is\Nsimilar to triangle CBA.
Dialogue: 0,0:05:25.13,0:05:27.42,Default,,0000,0000,0000,,And once again, we use this\Nexact same kind of argument
Dialogue: 0,0:05:27.42,0:05:28.44,Default,,0000,0000,0000,,that we did with this triangle.
Dialogue: 0,0:05:28.44,0:05:31.06,Default,,0000,0000,0000,,Well, if it's similar, the ratio\Nof all the corresponding sides
Dialogue: 0,0:05:31.06,0:05:32.07,Default,,0000,0000,0000,,have to be the same.
Dialogue: 0,0:05:32.07,0:05:33.38,Default,,0000,0000,0000,,And that ratio is 1/2.
Dialogue: 0,0:05:33.38,0:05:35.68,Default,,0000,0000,0000,,So the ratio of this\Nside to this side,
Dialogue: 0,0:05:35.68,0:05:41.42,Default,,0000,0000,0000,,the ratio of FD to\NAC, has to be 1/2.
Dialogue: 0,0:05:41.42,0:05:44.10,Default,,0000,0000,0000,,Or FD has to be 1/2 of AC.
Dialogue: 0,0:05:44.10,0:05:46.75,Default,,0000,0000,0000,,And 1/2 of AC is just\Nthe length of AE.
Dialogue: 0,0:05:46.75,0:05:50.71,Default,,0000,0000,0000,,So that is just going to be\Nthat length right over there.
Dialogue: 0,0:05:50.71,0:05:52.70,Default,,0000,0000,0000,,I think you see\Nwhere this is going.
Dialogue: 0,0:05:52.70,0:05:55.47,Default,,0000,0000,0000,,And also, because it's similar,\Nall of the corresponding angles
Dialogue: 0,0:05:55.47,0:05:56.49,Default,,0000,0000,0000,,have to be the same.
Dialogue: 0,0:05:56.49,0:05:58.99,Default,,0000,0000,0000,,And we know that\Nthe larger triangle
Dialogue: 0,0:05:58.99,0:06:00.87,Default,,0000,0000,0000,,has a yellow angle\Nright over there.
Dialogue: 0,0:06:00.87,0:06:03.77,Default,,0000,0000,0000,,So we'd have that yellow\Nangle right over here.
Dialogue: 0,0:06:03.77,0:06:05.14,Default,,0000,0000,0000,,And this triangle\Nright over here
Dialogue: 0,0:06:05.14,0:06:07.44,Default,,0000,0000,0000,,was also similar to\Nthe larger triangle.
Dialogue: 0,0:06:07.44,0:06:09.76,Default,,0000,0000,0000,,So it will have that same\Nangle measure up here.
Dialogue: 0,0:06:09.76,0:06:12.66,Default,,0000,0000,0000,,We already showed that\Nin this first part.
Dialogue: 0,0:06:12.66,0:06:14.86,Default,,0000,0000,0000,,So now let's go to\Nthis third triangle.
Dialogue: 0,0:06:14.86,0:06:16.03,Default,,0000,0000,0000,,I think you see the pattern.
Dialogue: 0,0:06:16.03,0:06:18.16,Default,,0000,0000,0000,,I'm sure you might be able\Nto just pause this video
Dialogue: 0,0:06:18.16,0:06:19.77,Default,,0000,0000,0000,,and prove it for yourself.
Dialogue: 0,0:06:19.77,0:06:25.80,Default,,0000,0000,0000,,But we see that the\Nratio of AF over AB
Dialogue: 0,0:06:25.80,0:06:28.78,Default,,0000,0000,0000,,is going to be the\Nsame as the ratio of AE
Dialogue: 0,0:06:28.78,0:06:33.38,Default,,0000,0000,0000,,over AC, which is equal to 1/2.
Dialogue: 0,0:06:33.38,0:06:36.97,Default,,0000,0000,0000,,So we have two corresponding\Nsides where the ratio is 1/2,
Dialogue: 0,0:06:36.97,0:06:38.96,Default,,0000,0000,0000,,from the smaller\Nto larger triangle.
Dialogue: 0,0:06:38.96,0:06:41.51,Default,,0000,0000,0000,,And they share a common angle.
Dialogue: 0,0:06:41.51,0:06:44.45,Default,,0000,0000,0000,,They share this angle in\Nbetween the two sides.
Dialogue: 0,0:06:44.45,0:06:47.47,Default,,0000,0000,0000,,So by SAS similarity--\Nthis is getting repetitive
Dialogue: 0,0:06:47.47,0:06:59.56,Default,,0000,0000,0000,,now-- we know that triangle\NEFA is similar to triangle CBA.
Dialogue: 0,0:07:03.28,0:07:05.46,Default,,0000,0000,0000,,And so the ratio of all\Nof the corresponding sides
Dialogue: 0,0:07:05.46,0:07:06.68,Default,,0000,0000,0000,,need to be 1/2.
Dialogue: 0,0:07:06.68,0:07:10.00,Default,,0000,0000,0000,,So the ratio of FE to\NBC needs to be 1/2,
Dialogue: 0,0:07:10.00,0:07:13.63,Default,,0000,0000,0000,,or FE needs to be 1/2 of that,\Nwhich is just the length of BD.
Dialogue: 0,0:07:13.63,0:07:17.72,Default,,0000,0000,0000,,So this is just going to be\Nthat length right over there.
Dialogue: 0,0:07:17.72,0:07:19.50,Default,,0000,0000,0000,,And you can also\Nsay that since we've
Dialogue: 0,0:07:19.50,0:07:22.38,Default,,0000,0000,0000,,shown that this triangle, this\Ntriangle, and this triangle--
Dialogue: 0,0:07:22.38,0:07:24.00,Default,,0000,0000,0000,,we haven't talked\Nabout this middle one
Dialogue: 0,0:07:24.00,0:07:26.64,Default,,0000,0000,0000,,yet-- they're all similar\Nto the larger triangle.
Dialogue: 0,0:07:26.64,0:07:28.89,Default,,0000,0000,0000,,So they're also all going\Nto be similar to each other.
Dialogue: 0,0:07:28.89,0:07:32.08,Default,,0000,0000,0000,,So they're all going to have\Nthe same corresponding angles.
Dialogue: 0,0:07:32.08,0:07:34.33,Default,,0000,0000,0000,,So if the larger triangle\Nhad this yellow angle here,
Dialogue: 0,0:07:34.33,0:07:35.79,Default,,0000,0000,0000,,then all of the\Ntriangles are going
Dialogue: 0,0:07:35.79,0:07:38.52,Default,,0000,0000,0000,,to have this yellow\Nangle right over there.
Dialogue: 0,0:07:38.52,0:07:40.62,Default,,0000,0000,0000,,And if the larger triangle\Nhad this blue angle
Dialogue: 0,0:07:40.62,0:07:43.28,Default,,0000,0000,0000,,right over here, then in\Nthe corresponding vertex,
Dialogue: 0,0:07:43.28,0:07:46.00,Default,,0000,0000,0000,,all of the triangles are\Ngoing to have that blue angle.
Dialogue: 0,0:07:46.00,0:07:47.88,Default,,0000,0000,0000,,All of the ones that\Nwe've shown are similar.
Dialogue: 0,0:07:47.88,0:07:50.75,Default,,0000,0000,0000,,We haven't thought about this\Nmiddle triangle just yet.
Dialogue: 0,0:07:50.75,0:07:52.78,Default,,0000,0000,0000,,And of course, if this\Nis similar to the whole,
Dialogue: 0,0:07:52.78,0:07:56.31,Default,,0000,0000,0000,,it'll also have this\Nangle at this vertex
Dialogue: 0,0:07:56.31,0:07:58.82,Default,,0000,0000,0000,,right over here, because this\Ncorresponds to that vertex,
Dialogue: 0,0:07:58.82,0:08:01.15,Default,,0000,0000,0000,,based on the similarity.
Dialogue: 0,0:08:01.15,0:08:02.07,Default,,0000,0000,0000,,So that's interesting.
Dialogue: 0,0:08:02.07,0:08:06.64,Default,,0000,0000,0000,,Now let's compare the\Ntriangles to each other.
Dialogue: 0,0:08:06.64,0:08:09.24,Default,,0000,0000,0000,,We've now shown that\Nall of these triangles
Dialogue: 0,0:08:09.24,0:08:11.20,Default,,0000,0000,0000,,have the exact same three sides.
Dialogue: 0,0:08:11.20,0:08:14.52,Default,,0000,0000,0000,,Has this blue side-- or\Nactually, this one-mark side,
Dialogue: 0,0:08:14.52,0:08:16.76,Default,,0000,0000,0000,,this two-mark side, and\Nthis three-mark side.
Dialogue: 0,0:08:16.76,0:08:18.22,Default,,0000,0000,0000,,One mark, two mark, three mark.
Dialogue: 0,0:08:18.22,0:08:19.91,Default,,0000,0000,0000,,One mark, two mark, three mark.
Dialogue: 0,0:08:19.91,0:08:22.67,Default,,0000,0000,0000,,And that even applies\Nto this middle triangle
Dialogue: 0,0:08:22.67,0:08:24.02,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:08:24.02,0:08:35.05,Default,,0000,0000,0000,,So by side-side-side\Ncongruency, we now know--
Dialogue: 0,0:08:35.05,0:08:38.00,Default,,0000,0000,0000,,and we want to be careful to get\Nour corresponding sides right--
Dialogue: 0,0:08:38.00,0:08:45.22,Default,,0000,0000,0000,,we now know that triangle CDE\Nis congruent to triangle DBF.
Dialogue: 0,0:08:49.71,0:08:51.29,Default,,0000,0000,0000,,I want to get the\Ncorresponding sides.
Dialogue: 0,0:08:51.29,0:08:51.98,Default,,0000,0000,0000,,I'm looking at the colors.
Dialogue: 0,0:08:51.98,0:08:54.42,Default,,0000,0000,0000,,I went from yellow to magenta\Nto blue, yellow, magenta,
Dialogue: 0,0:08:54.42,0:08:58.91,Default,,0000,0000,0000,,to blue, which is going to\Nbe congruent to triangle
Dialogue: 0,0:08:58.91,0:09:04.92,Default,,0000,0000,0000,,EFA, which is going to be\Ncongruent to this triangle
Dialogue: 0,0:09:04.92,0:09:05.42,Default,,0000,0000,0000,,in here.
Dialogue: 0,0:09:05.42,0:09:07.18,Default,,0000,0000,0000,,But we want to make\Nsure that we're
Dialogue: 0,0:09:07.18,0:09:09.48,Default,,0000,0000,0000,,getting the right\Ncorresponding sides here.
Dialogue: 0,0:09:09.48,0:09:11.01,Default,,0000,0000,0000,,So to make sure we\Ndo that, we just
Dialogue: 0,0:09:11.01,0:09:12.77,Default,,0000,0000,0000,,have to think about the angles.
Dialogue: 0,0:09:12.77,0:09:15.10,Default,,0000,0000,0000,,So we know-- and\Nthis is interesting--
Dialogue: 0,0:09:15.10,0:09:18.33,Default,,0000,0000,0000,,that because the interior\Nangles of a triangle
Dialogue: 0,0:09:18.33,0:09:20.82,Default,,0000,0000,0000,,add up to 180 degrees,\Nwe know this magenta
Dialogue: 0,0:09:20.82,0:09:24.60,Default,,0000,0000,0000,,angle plus this blue angle plus\Nthis yellow angle equal 180.
Dialogue: 0,0:09:24.60,0:09:27.01,Default,,0000,0000,0000,,Here, we have the blue\Nangle and the magenta angle,
Dialogue: 0,0:09:27.01,0:09:29.61,Default,,0000,0000,0000,,and clearly they will\Nall add up to 180.
Dialogue: 0,0:09:29.61,0:09:31.60,Default,,0000,0000,0000,,So you must have the blue angle.
Dialogue: 0,0:09:31.60,0:09:33.85,Default,,0000,0000,0000,,The blue angle must\Nbe right over here.
Dialogue: 0,0:09:33.85,0:09:37.30,Default,,0000,0000,0000,,Same argument-- yellow\Nangle and blue angle,
Dialogue: 0,0:09:37.30,0:09:39.51,Default,,0000,0000,0000,,we must have the magenta\Nangle right over here.
Dialogue: 0,0:09:39.51,0:09:40.69,Default,,0000,0000,0000,,They add up to 180.
Dialogue: 0,0:09:40.69,0:09:42.25,Default,,0000,0000,0000,,So this must be\Nthe magenta angle.
Dialogue: 0,0:09:42.25,0:09:44.63,Default,,0000,0000,0000,,And then finally,\Nmagenta and blue-- this
Dialogue: 0,0:09:44.63,0:09:47.19,Default,,0000,0000,0000,,must be the yellow\Nangle right over there.
Dialogue: 0,0:09:47.19,0:09:48.90,Default,,0000,0000,0000,,And so when we wrote\Nthe congruency here,
Dialogue: 0,0:09:48.90,0:09:50.89,Default,,0000,0000,0000,,we started at CDE.
Dialogue: 0,0:09:50.89,0:09:53.71,Default,,0000,0000,0000,,We went yellow, magenta, blue.
Dialogue: 0,0:09:53.71,0:09:57.19,Default,,0000,0000,0000,,So over here, we're going\Nto go yellow, magenta, blue.
Dialogue: 0,0:09:57.19,0:09:59.54,Default,,0000,0000,0000,,So it's going to be\Ncongruent to triangle FED.
Dialogue: 0,0:10:05.44,0:10:06.62,Default,,0000,0000,0000,,And so that's pretty cool.
Dialogue: 0,0:10:06.62,0:10:09.47,Default,,0000,0000,0000,,We just showed that all\Nthree, that this triangle,
Dialogue: 0,0:10:09.47,0:10:11.43,Default,,0000,0000,0000,,this triangle, this\Ntriangle, and that triangle
Dialogue: 0,0:10:11.43,0:10:12.53,Default,,0000,0000,0000,,are congruent.
Dialogue: 0,0:10:12.53,0:10:14.83,Default,,0000,0000,0000,,And also, we can look\Nat the corresponding--
Dialogue: 0,0:10:14.83,0:10:17.68,Default,,0000,0000,0000,,and that they all have\Nratios relative to-- they're
Dialogue: 0,0:10:17.68,0:10:20.52,Default,,0000,0000,0000,,all similar to the larger\Ntriangle, to triangle ABC.
Dialogue: 0,0:10:20.52,0:10:23.43,Default,,0000,0000,0000,,And that the ratio between\Nthe sides is 1 to 2.
Dialogue: 0,0:10:23.43,0:10:25.94,Default,,0000,0000,0000,,And also, because we've looked\Nat corresponding angles,
Dialogue: 0,0:10:25.94,0:10:30.52,Default,,0000,0000,0000,,we see, for example,\Nthat this angle
Dialogue: 0,0:10:30.52,0:10:31.77,Default,,0000,0000,0000,,is the same as that angle.
Dialogue: 0,0:10:31.77,0:10:33.86,Default,,0000,0000,0000,,So if you viewed DC\Nor if you viewed BC
Dialogue: 0,0:10:33.86,0:10:35.71,Default,,0000,0000,0000,,as a transversal,\Nall of a sudden
Dialogue: 0,0:10:35.71,0:10:40.44,Default,,0000,0000,0000,,it becomes pretty clear that FD\Nis going to be parallel to AC,
Dialogue: 0,0:10:40.44,0:10:43.13,Default,,0000,0000,0000,,because the corresponding\Nangles are congruent.
Dialogue: 0,0:10:43.13,0:10:46.90,Default,,0000,0000,0000,,So this is going to be parallel\Nto that right over there.
Dialogue: 0,0:10:46.90,0:10:49.06,Default,,0000,0000,0000,,And then you could use\Nthat same exact argument
Dialogue: 0,0:10:49.06,0:10:51.78,Default,,0000,0000,0000,,to say, well, then this\Nside, because once again,
Dialogue: 0,0:10:51.78,0:10:56.08,Default,,0000,0000,0000,,corresponding angles\Nhere and here-- you
Dialogue: 0,0:10:56.08,0:10:58.49,Default,,0000,0000,0000,,could say that\Nthis is going to be
Dialogue: 0,0:10:58.49,0:11:01.75,Default,,0000,0000,0000,,parallel to that\Nright over there.
Dialogue: 0,0:11:01.75,0:11:06.36,Default,,0000,0000,0000,,And then finally, you make\Nthe same argument over here.
Dialogue: 0,0:11:06.36,0:11:08.82,Default,,0000,0000,0000,,I want to make sure I get the\Nright corresponding angles.
Dialogue: 0,0:11:08.82,0:11:12.06,Default,,0000,0000,0000,,You have this line\Nand this line.
Dialogue: 0,0:11:12.06,0:11:13.85,Default,,0000,0000,0000,,And this angle\Ncorresponds to that angle.
Dialogue: 0,0:11:13.85,0:11:14.98,Default,,0000,0000,0000,,They're the same.
Dialogue: 0,0:11:14.98,0:11:21.41,Default,,0000,0000,0000,,So this DE must\Nbe parallel to BA.
Dialogue: 0,0:11:21.41,0:11:25.28,Default,,0000,0000,0000,,So that's another neat property\Nof this medial triangle,
Dialogue: 0,0:11:25.28,0:11:27.41,Default,,0000,0000,0000,,[? I thought. ?]\NAll of these things
Dialogue: 0,0:11:27.41,0:11:29.95,Default,,0000,0000,0000,,just jump out when you just try\Nto do something fairly simple
Dialogue: 0,0:11:29.95,0:11:32.18,Default,,0000,0000,0000,,with a triangle.