[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.20,Default,,0000,0000,0000,,What I wanna do on this video, for some triangle we're gonna focused
Dialogue: 0,0:00:04.21,0:00:07.68,Default,,0000,0000,0000,,on this larger triangle over heretriangle ABC
Dialogue: 0,0:00:07.69,0:00:11.69,Default,,0000,0000,0000,,What I wanna do is prove that circumcenter,
Dialogue: 0,0:00:11.70,0:00:15.02,Default,,0000,0000,0000,,remember the circumcenter is the intersection
Dialogue: 0,0:00:15.03,0:00:16.82,Default,,0000,0000,0000,,of it's perpendicular bisectors
Dialogue: 0,0:00:16.83,0:00:21.92,Default,,0000,0000,0000,,The circumcenter for this triangle, the centroid of this triangle,
Dialogue: 0,0:00:21.93,0:00:24.04,Default,,0000,0000,0000,,the centroid is the intersection of it's medians,
Dialogue: 0,0:00:24.05,0:00:28.72,Default,,0000,0000,0000,,and the orthocenter, that's the intersection of it's altitudes
Dialogue: 0,0:00:28.73,0:00:35.46,Default,,0000,0000,0000,,all sit on the same line, or that OI right over here really a line segment
Dialogue: 0,0:00:35.47,0:00:39.20,Default,,0000,0000,0000,,Or that OG and GI are really just 2 segments
Dialogue: 0,0:00:39.21,0:00:44.81,Default,,0000,0000,0000,,that make up these 2 line segments which is part of the euler line
Dialogue: 0,0:00:44.82,0:00:48.90,Default,,0000,0000,0000,,And to do that, I've set up a medial triangle right over here,
Dialogue: 0,0:00:48.91,0:00:53.13,Default,,0000,0000,0000,,triangle FED, or actually I should say, triangle DEF,
Dialogue: 0,0:00:53.14,0:00:56.76,Default,,0000,0000,0000,,whch is the medial triangle for ABC
Dialogue: 0,0:00:56.77,0:00:58.67,Default,,0000,0000,0000,,And there's already a bunch of things
Dialogue: 0,0:00:58.68,0:01:00.67,Default,,0000,0000,0000,,that we know about medial triangle
Dialogue: 0,0:01:00.68,0:01:02.64,Default,,0000,0000,0000,,and we've proven this in previous videos
Dialogue: 0,0:01:02.65,0:01:07.01,Default,,0000,0000,0000,,One thing we know, is the medial triangle DEF,
Dialogue: 0,0:01:07.04,0:01:11.30,Default,,0000,0000,0000,,DEF is going to be similar to the larger triangle
Dialogue: 0,0:01:11.31,0:01:13.41,Default,,0000,0000,0000,,the triangle that is a medial triangle of,
Dialogue: 0,0:01:13.42,0:01:18.00,Default,,0000,0000,0000,,it is of, and ratio from the larger triangle to the smaller triangle
Dialogue: 0,0:01:18.01,0:01:19.76,Default,,0000,0000,0000,,it's a 2 to 1 ratio
Dialogue: 0,0:01:19.77,0:01:21.78,Default,,0000,0000,0000,,And this is really important to proof
Dialogue: 0,0:01:21.79,0:01:24.63,Default,,0000,0000,0000,,When two triangles are similar with the given ratio,
Dialogue: 0,0:01:24.64,0:01:28.36,Default,,0000,0000,0000,,that means if you take the distance between any 2 corresponding parts
Dialogue: 0,0:01:28.37,0:01:32.69,Default,,0000,0000,0000,,of the two similar triangles that ratio will be 2 to 1
Dialogue: 0,0:01:32.70,0:01:35.92,Default,,0000,0000,0000,,Now the other relationship that we've already shown,
Dialogue: 0,0:01:35.93,0:01:38.35,Default,,0000,0000,0000,,the other relationship between the medial triangle,
Dialogue: 0,0:01:38.36,0:01:40.44,Default,,0000,0000,0000,,and the triangle that is the medial triangle
Dialogue: 0,0:01:40.45,0:01:43.59,Default,,0000,0000,0000,,of is that we've shown the orthocenter
Dialogue: 0,0:01:43.60,0:01:52.13,Default,,0000,0000,0000,,of the medial center of the larger triangle
Dialogue: 0,0:01:52.14,0:01:55.11,Default,,0000,0000,0000,,So one way to think about it is Point O,
Dialogue: 0,0:01:55.12,0:01:58.81,Default,,0000,0000,0000,,we already mentioned is the circumcenter of the larger triangle
Dialogue: 0,0:01:58.82,0:02:05.56,Default,,0000,0000,0000,,It is also the, it is also the orthocenter of the smaller triangle,
Dialogue: 0,0:02:05.57,0:02:06.88,Default,,0000,0000,0000,,we actually wrote it of here
Dialogue: 0,0:02:06.89,0:02:12.11,Default,,0000,0000,0000,,so Point O noticed it is on this perpendicular bisector,
Dialogue: 0,0:02:12.12,0:02:14.88,Default,,0000,0000,0000,,you know, I should do a bunch of other ones of this dark grey color
Dialogue: 0,0:02:14.89,0:02:18.64,Default,,0000,0000,0000,,but I didn't wanna make this diagram too messy
Dialogue: 0,0:02:18.65,0:02:21.22,Default,,0000,0000,0000,,But this is the circumcenter of the larger triangle
Dialogue: 0,0:02:57.12,0:03:04.56,Default,,0000,0000,0000,,Now in order to prove that OG and I all sit on the same line
Dialogue: 0,0:03:04.57,0:03:07.03,Default,,0000,0000,0000,,or the same segment in this case
Dialogue: 0,0:03:07.04,0:03:10.86,Default,,0000,0000,0000,,What I'm going to do, I wanna prove,
Dialogue: 0,0:03:10.87,0:03:16.10,Default,,0000,0000,0000,,I'm going to prove that triangle FOG,
Dialogue: 0,0:03:16.11,0:03:19.36,Default,,0000,0000,0000,,I'm going to prove that triangle FOG,
Dialogue: 0,0:03:19.37,0:03:25.42,Default,,0000,0000,0000,,is similar to, is similar to triangle CIG,
Dialogue: 0,0:03:25.43,0:03:29.18,Default,,0000,0000,0000,,is similar to triangle CIG
Dialogue: 0,0:03:29.19,0:03:32.09,Default,,0000,0000,0000,,Because if I can prove that, then their corresponding angles
Dialogue: 0,0:03:32.10,0:03:33.29,Default,,0000,0000,0000,,are going to be equivalent,
Dialogue: 0,0:03:33.30,0:03:35.69,Default,,0000,0000,0000,,you could say this is angle is going to be equal
Dialogue: 0,0:03:35.70,0:03:36.94,Default,,0000,0000,0000,,to those angle over here
Dialogue: 0,0:03:36.95,0:03:39.76,Default,,0000,0000,0000,,And so OI would have to be a transversal,
Dialogue: 0,0:03:39.77,0:03:42.27,Default,,0000,0000,0000,,cause we're goiong to see these two lines here are parallel
Dialogue: 0,0:03:42.28,0:03:45.39,Default,,0000,0000,0000,,or if these two triangles are similar,
Dialogue: 0,0:03:45.40,0:03:47.83,Default,,0000,0000,0000,,just remember that someone's look at our triangle here
Dialogue: 0,0:03:47.84,0:03:49.33,Default,,0000,0000,0000,,and this triangles over there
Dialogue: 0,0:03:49.34,0:03:51.40,Default,,0000,0000,0000,,If they truly are similar then this angle
Dialogue: 0,0:03:51.41,0:03:52.78,Default,,0000,0000,0000,,is going to be equal to that angle
Dialogue: 0,0:03:52.79,0:03:53.74,Default,,0000,0000,0000,,which would mean that,
Dialogue: 0,0:03:53.75,0:03:56.18,Default,,0000,0000,0000,,so these are really would be vertical angles
Dialogue: 0,0:03:56.19,0:03:58.97,Default,,0000,0000,0000,,and so this really would be real line
Dialogue: 0,0:03:58.98,0:04:01.27,Default,,0000,0000,0000,,So let's go the actual proof
Dialogue: 0,0:04:01.28,0:04:05.55,Default,,0000,0000,0000,,So maybe, I don't need those to highlight it over here
Dialogue: 0,0:04:05.56,0:04:07.62,Default,,0000,0000,0000,,So one thing and I've hinted about this already,
Dialogue: 0,0:04:07.63,0:04:12.39,Default,,0000,0000,0000,,we know that this line over here, we can call this like XC
Dialogue: 0,0:04:12.40,0:04:15.58,Default,,0000,0000,0000,,we know this is perpendicular like AB
Dialogue: 0,0:04:15.59,0:04:20.29,Default,,0000,0000,0000,,it is an altitude, and we also know that FY right over here
Dialogue: 0,0:04:20.30,0:04:25.32,Default,,0000,0000,0000,,is perpendicular to AB, it is a perpendicular bi-sector
Dialogue: 0,0:04:25.33,0:04:28.39,Default,,0000,0000,0000,,So they both form the same angle with a transversal
Dialogue: 0,0:04:28.40,0:04:29.95,Default,,0000,0000,0000,,you can view AB as a transversal
Dialogue: 0,0:04:29.96,0:04:33.24,Default,,0000,0000,0000,,so they must be parallel, so we know that FY,
Dialogue: 0,0:04:33.25,0:04:39.11,Default,,0000,0000,0000,,FY is parallel to XC, to XC
Dialogue: 0,0:04:39.12,0:04:39.35,Default,,0000,0000,0000,,segment FY is parallel to XC,
Dialogue: 0,0:04:39.36,0:04:41.98,Default,,0000,0000,0000,,segment FY is parallel to segment XC
Dialogue: 0,0:04:41.99,0:04:46.74,Default,,0000,0000,0000,,And we can write it; this guy is parallel to that guy there
Dialogue: 0,0:04:46.75,0:04:51.49,Default,,0000,0000,0000,,And that's useful because we know that alternate interior angles
Dialogue: 0,0:04:51.50,0:04:54.70,Default,,0000,0000,0000,,of a transversal, when a transversal intersects two parallel
Dialogue: 0,0:04:54.71,0:04:56.07,Default,,0000,0000,0000,,lines are congruent
Dialogue: 0,0:04:56.08,0:05:02.67,Default,,0000,0000,0000,,So we know, we know that this angle, so we know that FC is a line
Dialogue: 0,0:05:02.68,0:05:06.74,Default,,0000,0000,0000,,it is a median of this larger triangle, triangle ABC
Dialogue: 0,0:05:06.90,0:05:09.90,Default,,0000,0000,0000,,So you have a line intersecting two parallel lines,
Dialogue: 0,0:05:09.91,0:05:12.86,Default,,0000,0000,0000,,alternate interior angels are congruent
Dialogue: 0,0:05:12.87,0:05:15.89,Default,,0000,0000,0000,,So that angle is gonna be congruent to that angle
Dialogue: 0,0:05:15.90,0:05:24.16,Default,,0000,0000,0000,,So you could say angle OFG is congruent to angle,
Dialogue: 0,0:05:24.17,0:05:27.72,Default,,0000,0000,0000,,so it's OFG, it's congruent to angel ICG,
Dialogue: 0,0:05:27.73,0:05:34.14,Default,,0000,0000,0000,,to ICG, now the other, the other thing we know,
Dialogue: 0,0:05:34.15,0:05:38.28,Default,,0000,0000,0000,,this is a property, this is a property of medians is that
Dialogue: 0,0:05:38.29,0:05:41.91,Default,,0000,0000,0000,,a median splits up or should I say the centroid,
Dialogue: 0,0:05:41.92,0:05:45.69,Default,,0000,0000,0000,,splits the median in to two segments that have a ratio of two to one
Dialogue: 0,0:05:45.70,0:05:47.36,Default,,0000,0000,0000,,or another way to think about this is a centroid
Dialogue: 0,0:05:47.37,0:05:50.83,Default,,0000,0000,0000,,is two thirds along the median
Dialogue: 0,0:05:50.84,0:05:53.59,Default,,0000,0000,0000,,So we know we've proven this on a previous video
Dialogue: 0,0:05:53.60,0:06:02.33,Default,,0000,0000,0000,,We know that CG, CG is a equal to 2 times GF, 2 times GF
Dialogue: 0,0:06:02.34,0:06:03.61,Default,,0000,0000,0000,,and I think you see where you are going,
Dialogue: 0,0:06:03.62,0:06:06.71,Default,,0000,0000,0000,,we have an angle, I've shown you that the ratio of this side
Dialogue: 0,0:06:06.72,0:06:09.48,Default,,0000,0000,0000,,to this side is 2 to 1 and that's just the property
Dialogue: 0,0:06:09.49,0:06:10.83,Default,,0000,0000,0000,,of centroids and medians
Dialogue: 0,0:06:10.84,0:06:16.85,Default,,0000,0000,0000,,And now if we can show you the ratio of his side CI is FO is 2 to 1
Dialogue: 0,0:06:16.86,0:06:20.24,Default,,0000,0000,0000,,that we have two corresponding sides where the ratio is 2 to 1,
Dialogue: 0,0:06:20.25,0:06:22.10,Default,,0000,0000,0000,,and we have the angle and between this congruent,
Dialogue: 0,0:06:22.11,0:06:25.92,Default,,0000,0000,0000,,and we have the SAS singularity to show that these two triangles
Dialogue: 0,0:06:25.93,0:06:28.57,Default,,0000,0000,0000,,are actually similar, so let's actually think about that
Dialogue: 0,0:06:28.58,0:06:34.18,Default,,0000,0000,0000,,CI is the distance between, CI is the between the larger triangle's
Dialogue: 0,0:06:34.19,0:06:40.21,Default,,0000,0000,0000,,point C orthocenter of the larger triangle
Dialogue: 0,0:06:40.22,0:06:41.83,Default,,0000,0000,0000,,Well what is FO?
Dialogue: 0,0:06:41.84,0:06:47.22,Default,,0000,0000,0000,,Well F is a corresponding point to point C on the medial triangle
Dialogue: 0,0:06:47.23,0:06:51.63,Default,,0000,0000,0000,,and we make sure that we specify the similarity with the right, F,
Dialogue: 0,0:06:51.64,0:06:54.04,Default,,0000,0000,0000,,F corresponds to point C
Dialogue: 0,0:06:54.05,0:06:59.37,Default,,0000,0000,0000,,So FO is the distance between F on the smaller medial triangle
Dialogue: 0,0:06:59.38,0:07:03.13,Default,,0000,0000,0000,,and the smaller medial triangle's orthocenter
Dialogue: 0,0:07:03.14,0:07:04.81,Default,,0000,0000,0000,,So this is the distance between C
Dialogue: 0,0:07:04.82,0:07:06.74,Default,,0000,0000,0000,,and the orthocenter of the larger triangle
Dialogue: 0,0:07:06.75,0:07:09.56,Default,,0000,0000,0000,,This is the distance between the corresponding
Dialogue: 0,0:07:09.57,0:07:10.99,Default,,0000,0000,0000,,side of the medial triangle
Dialogue: 0,0:07:11.00,0:07:13.07,Default,,0000,0000,0000,,and the smaller medial triangle and it's orthocenter
Dialogue: 0,0:07:13.08,0:07:16.19,Default,,0000,0000,0000,,So this is the same corresponding distance
Dialogue: 0,0:07:16.20,0:07:18.74,Default,,0000,0000,0000,,on the larger triangle and the medial triangle,
Dialogue: 0,0:07:18.75,0:07:21.82,Default,,0000,0000,0000,,and we already know that they're similar with the ratio of 2 to 1
Dialogue: 0,0:07:21.83,0:07:25.82,Default,,0000,0000,0000,,And so the corresponding distances between any 2 points
Dialogue: 0,0:07:25.83,0:07:28.56,Default,,0000,0000,0000,,on the two same triangle are gonna have the same ratio
Dialogue: 0,0:07:28.57,0:07:32.75,Default,,0000,0000,0000,,So because of that similarity, because of the similarity
Dialogue: 0,0:07:32.76,0:07:39.50,Default,,0000,0000,0000,,we know that CI, CI is gonna be equal to 2 times FO
Dialogue: 0,0:07:39.51,0:07:43.44,Default,,0000,0000,0000,,I wanna emphasize this; C is the corresponding point to F,
Dialogue: 0,0:07:43.45,0:07:46.08,Default,,0000,0000,0000,,when we look at both of these similar triangles,
Dialogue: 0,0:07:46.09,0:07:48.60,Default,,0000,0000,0000,,I is the orthocenter of the larger triangle,
Dialogue: 0,0:07:48.61,0:07:50.56,Default,,0000,0000,0000,,O is the orthocenter of the smaller triangle
Dialogue: 0,0:07:50.57,0:07:52.47,Default,,0000,0000,0000,,You're taking a corresponding point
Dialogue: 0,0:07:52.48,0:07:54.27,Default,,0000,0000,0000,,to the orthocenter of the larger triangle,
Dialogue: 0,0:07:54.28,0:07:58.32,Default,,0000,0000,0000,,corresponding point of the smaller triangle
Dialogue: 0,0:07:58.33,0:08:00.71,Default,,0000,0000,0000,,The triangles are similar to the ratios of 2 to 1
Dialogue: 0,0:08:00.72,0:08:04.93,Default,,0000,0000,0000,,so the ratio of this length to this length is going to be 2 to 1
Dialogue: 0,0:08:04.94,0:08:07.91,Default,,0000,0000,0000,,So we've shown, we've shown the ratio
Dialogue: 0,0:08:07.92,0:08:12.38,Default,,0000,0000,0000,,of this side to this side is 2 to 1
Dialogue: 0,0:08:12.45,0:08:17.73,Default,,0000,0000,0000,,We've shown the ratio if this side to side is also to 2 to 1
Dialogue: 0,0:08:17.74,0:08:20.37,Default,,0000,0000,0000,,We've shown the angle in between are,
Dialogue: 0,0:08:20.38,0:08:24.08,Default,,0000,0000,0000,,the angle between them is congruent
Dialogue: 0,0:08:24.09,0:08:26.95,Default,,0000,0000,0000,,So we have proven by SAS similarity,
Dialogue: 0,0:08:26.96,0:08:33.54,Default,,0000,0000,0000,,so it goes down a little bit, so by, by SAS similarity,
Dialogue: 0,0:08:33.55,0:08:40.11,Default,,0000,0000,0000,,not congruency, similarity we've proven that triangle FOG
Dialogue: 0,0:08:40.12,0:08:43.36,Default,,0000,0000,0000,,is similar to CIG
Dialogue: 0,0:08:43.37,0:08:46.35,Default,,0000,0000,0000,,And so we know corresponding triangles are congruent,
Dialogue: 0,0:08:46.36,0:08:51.91,Default,,0000,0000,0000,,we know that angle CIG correspond to angle FOG
Dialogue: 0,0:08:51.92,0:08:57.45,Default,,0000,0000,0000,,so those are going to be congruent, and we also know that angle CGI,
Dialogue: 0,0:08:57.46,0:09:00.66,Default,,0000,0000,0000,,angle CGI, let me do this a new color,
Dialogue: 0,0:09:00.67,0:09:04.46,Default,,0000,0000,0000,,angle CGI corresponds to angel OGF
Dialogue: 0,0:09:04.47,0:09:07.27,Default,,0000,0000,0000,,so they're also going to be congruent
Dialogue: 0,0:09:07.28,0:09:09.36,Default,,0000,0000,0000,,So you can look at the different ways of these angle
Dialogue: 0,0:09:09.37,0:09:12.14,Default,,0000,0000,0000,,and this angle are the same you can now view OI
Dialogue: 0,0:09:12.15,0:09:15.67,Default,,0000,0000,0000,,as a true line as a transversal of these two parallel lines
Dialogue: 0,0:09:15.68,0:09:17.39,Default,,0000,0000,0000,,So that let's you know that's a one line
Dialogue: 0,0:09:17.40,0:09:18.92,Default,,0000,0000,0000,,or you can look these two over here,
Dialogue: 0,0:09:18.93,0:09:21.24,Default,,0000,0000,0000,,so look these two angles are equivalent
Dialogue: 0,0:09:21.25,0:09:23.07,Default,,0000,0000,0000,,so these must be vertical angles,
Dialogue: 0,0:09:23.08,0:09:26.80,Default,,0000,0000,0000,,so this must actually be, this actually must be the same line
Dialogue: 0,0:09:26.81,0:09:29.29,Default,,0000,0000,0000,,The angle that this is approaching, this,
Dialogue: 0,0:09:29.30,0:09:32.29,Default,,0000,0000,0000,,this median is the same angle that is leaving
Dialogue: 0,0:09:32.30,0:09:35.84,Default,,0000,0000,0000,,So these are all, these are definitely on the same line
Dialogue: 0,0:09:35.85,0:09:38.31,Default,,0000,0000,0000,,So it's a very simple proof, once again,
Dialogue: 0,0:09:38.32,0:09:42.26,Default,,0000,0000,0000,,from a very profound idea, the orthocenter,
Dialogue: 0,0:09:42.27,0:09:45.93,Default,,0000,0000,0000,,the centroid and the median of any triangle
Dialogue: 0,0:09:45.94,0:09:49.32,Default,,0000,0000,0000,,all sit on this magical euler's line