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Current time:0:00Total duration:9:50

Video transcript

what I want to do in this video is that for some triangle we're going to focus on this larger triangle over here triangle ABC what I want to do is prove that the circumcenter of this triangle remember the circumcenter is the intersection of its perpendicular bisectors that the circumcenter for this triangle the centroid of this triangle the centroid is the intersection of its medians and the orthocenter of this triangle that's the intersection of its altitudes all sit on the same line or that o i right over here really is a line segment or that og and GI are really just two segments that make up this larger line segment which is part of the Euler line and to do that I've set up a medial triangle right over here triangle Fe D where I should I should say triangle d EF which is the medial triangle for a b c and there's already a bunch of things that we know about medial triangles and we've proven this in previous videos one thing we know is that the medial triangle a d EF d EF is going to be similar to the larger triangle the triangle it is a medial triangle of it is of and that ratio from the larger triangle to the smaller triangle is a two to one ratio and this is going to be really important to our proof when two triangles are similar with the given ratio that means that if you take the distance between any two corresponding parts of the two similar triangles that ratio will be two to one now the other relationship that we've already shown the other relationship between the medial triangle and the triangle it is the medial triangle of is that we've shown that the orthocenter of the medial triangle the orthocenter of the medial triangle is the circumcenter of the larger triangle so one way to think about it point O we already mentioned is the circumcenter of the larger triangle it is also the it is also the orthocenter of the smaller triangle and we actually wrote it up here so point O notice it is on this out it is on this perpendicular bisector over here and actually drew a bunch of other ones in the dark gray color I didn't want a cloud I didn't want to make this diagram too messy but this is the circumcenter of the larger triangle and it is also the orthocenter of the smaller triangle of d EF and we actually use this fact when we wanted to prove that ortho centers are concurrent we started with the medial triangle we said okay let's find its or let's let's think about where the altitudes intersect and we said well like all of these altitudes are actually perpendicular bisectors for a larger triangle if we assume that this is the medial triangle of that larger triangle so point O and this is going to be important to our proof it is the circumcenter of triangle ABC but it's the orthocenter of d EF and we've already talked about this in previous videos now in order to prove that og and I all sit on the same line or the same segment in this case what I'm going to do is prove I want to prove I'm going to prove that triangle F oh gee I'm going to prove the triangle f OG is similar is similar to triangle C I G is similar to triangle C I G because if I can prove that then their corresponding angles are going to be equivalent you could say that this angle is going to be equal to this angle over here and so oh I would have to be a transversal because we're going to see that these two lines over here parallel or if they if these two triangles are similar so remember we're looking at this triangle right over here and this triangle over there if they really are similar then this angle is going to be equal to that angle which would mean that they're so these are really would be vertical angles and so this really would be a line so let's go to the actual proof so maybe I well I won't leave those two highlighted right there so one thing and I hint at this already we know that this line right over here we could call this line XC we know this is perpendicular line a B it is an altitude and we also know that F Y right over here is perpendicular to a B it is a perpendicular bisector so they say they both form the same angle with a transversal you could view a B as a transversal so they must be parallel so we know that F Y F Y is parallel to X c2 X C segment F Y is parallel to segment X C and we could write it like this this this guy is parallel to that guy there and that's useful because we know that alternate interior angles of a transversal when a transversal intersects two parallel lines are congruent so we know we know that this angle so we know that FC is a is a line it is a median of this larger triangle triangle ABC so you have a line intersecting two parallel lines alternate interior angles are congruent so that angle is going to be congruent to that angle so we could say angle o FG angle o FG is congruent to angle so o FG is congruent to angle IC g - i see G now the other thing the other thing we know and this is a property this is a property of medians is that a median splits up or I should say the centroids splits the median into two segments that have a ratio of two to one or another way to think about is the centroid is two-thirds along along the median so we know we've proven this in a previous video we know that C G C G is equal to 2 times GF 2 times G F and I think you see where you're going we're going here we have an angle I've shown you that the ratio of this side to this side is 2 to 1 and that's just the property of centroids and medians and now if we can show you that the ratio of this side CI to fo is 2 to 1 then we have two corresponding sides where the ratio is 2 to 1 and we have the angle in between this congruent we could use SAS similarity to show that these two triangles are actually similar so let's actually think about that ci is the distance between CI is the distance between the larger triangles Point C and its orthocenter right eye is the orthocenter of the larger triangle well what is fo well F is a corresponding point to point C on the medial triangle and we we make sure that we specify the similarity with the right F f corresponds to point C so f o is the distance between F on the smaller medial triangle and the smaller medial triangles orthocenter so this is the distance between C and the orthocenter of the larger triangle this is this is the distance between the corresponding side of the medial triangle and its orthocenter so these this is the same corresponding distance on the larger triangle and on the medial triangle and we already know that they have they're similar with the ratio of 2 to 1 and so the corresponding distance is between any two points on the two triangles we're going to have the same ratio so because of that similarity because of that similarity we know that CI CI is going to be equal to 2 times fo I want to emphasize this C is the corresponding point to F when we look at both of these similar triangles eyes or at the centre of the larger triangle o is the orthocenter of the smaller triangle you're taking a corresponding point to the orthocenter of the larger triangle corresponding point of the smaller triangle to the orthocenter of the smaller triangle the triangles are similar in a ratio of 2 to 1 so the ratio of this length to this length is going to be 2 to 1 so we've shown we've shown that the ratio of this side to this side is 2 to 1 we've shown that the ratio of this side to this side is also 2 to 1 and we've shown that the angle in between them are the angle between in between them is congruent so we have proven by SAS similarity so let me scroll down a little bit so by by SAS similarity not congruence similarity we've proven that triangle f OG is similar to triangle c IG and so we know corresponding angles are congruent we know that angle c IG corresponds to angle fo G so those are going to be congruent and we also know that angle cgi angles cgi let me do this a new color angle CGI corresponds to angle o GF so there are also going to be congruent so you can look at at different ways of this angle and this angle are the same you can now have uoy as a true line as a transversal of these two parallel lines so that lets you know it's one line or you could look at these two over here say look these two angles are equivalent so these must be vertical angles and so this must actually be this actually must be the same line the angle that this is approaching this this median right over here is the same angle that it's leaving so these are all these are definitely on the same line so it's a very simple proof once again for a very profound idea that the orthocenter the centroid and the median of any triangle all sit on this magical Euler's line