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In this video, let's discuss something interesting about equilateral triangles, we have learned what

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we mean with median angle bisect perpendicular by sector and altitude, and we have also seen their

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respective meeting points in a triangle.

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All right.

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Now let's see those in the case of an equilateral triangle.

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Over here, we have an equilateral triangle led the side of this equilateral triangle.

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We have linked it now in an equilateral triangle.

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You will see that the median angle by sector, all the different perpendicular bassnectar are all the

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same line.

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Let's try to understand why.

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Or here I have a triangle right now.

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Imagine let this be the median.

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Let's just take this as the median.

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So if you observe this triangle over here and this triangle over here, you can see that they are congruent.

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Why would that be?

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This side is equal to decide model of length.

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If we have constructed this side over here as the median right there for this.

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And this will be equal because the median joins the vertex with the midpoint of the opposite side.

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But these two sides are also equal and this side is a common sight.

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Therefore, you will see that this triangle over here and this triangle over here are congruent right

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now because these triangles are congruent.

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What are the things that we can derive out of it?

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We can see that this angle.

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And this angle has to be equal, right, because there corresponding angles of two congruent triangles

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and you will see that these two angles form a linear bent.

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If I say that this is X and this is, again, X degree, we have seen they are equal now because they

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form a linear pair, X plus X is equal to 180 degrees.

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Therefore X is equal to 90 degrees.

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Right.

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We have seen that X will be equal to 90 degrees.

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And because X is equal to 90 degrees.

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Right.

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You can see that this line over here is the perpendicular by sector as well, because it's bisecting

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this side and it is perpendicular.

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So we have seen that the median and the perpendicular by sector are the same now because this triangle

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and this triangle is congruent.

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Therefore, this angle over here and this angle over here will be equal.

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Right, because again, there corresponding angles of two congruent triangles.

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Therefore, this line over here is also the angle by sector.

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Right.

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And because this angle over here is 90 degrees and it is passing through this vertex, it's also an

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altitude because we have learned that an altitude joins the vertex and it's a perpendicular drop to

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the opposite side through a vertex.

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So this and this side over here is also an altitude.

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Therefore, in an equilateral triangle, the median angle by sector altitude and perpendicular by sector

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are all the same line.

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All right.

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Now, having established this, let's move on to we have drawn one median over here.

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Similarly, let's draw the other two also and we will see that they intersect at this point.

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Let me name these points so it'll be easy for us.

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Let this point be point G now, because this line this line in this line is a median angled by altitude

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and perpendicular by sector.

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Therefore, point the G is the center, sock'em center, centroid and auto center.

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So in an equilateral triangle, all these points going side.

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So it's the same point.

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The center is also the subcommander, it's also the centroid and it's also the auto center.

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Now, let me mark these points, SPQR Now we know that in the case of a centroid, right, it divides

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the median in the ratio, two is to one.

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So this is very useful, especially in the case of an equilateral triangle in a lot of problems.

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Right.

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Because we have seen that in an equal triangle.

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All these four are the same line.

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Right.

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Therefore, I can see that over here.

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Let's take, for example, this line a bit so I can see it divided by g.p is equal to two by one, because

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this is also the centroid.

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And we know that the centroid divides of media in the ratio two is to one, therefore aided by GPS equals

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two by one.

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Now let this be the height of the triangle.

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We know that this angle is 90 degrees, right?

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Therefore, AP is also an altitude.

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So therefore AP is the height from point A to B, B.C., let's call it as it's now.

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We know that in an equal triangle, we have seen in a previous video that it is equal to Route three

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by two it.

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Where is the side of the equilateral triangle?

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All right.

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Now we've seen that divides up in the ratio two by one.

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Right?

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So I can say that a G is equal to two by three times its right.

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This is two weeks and this is X-Rite, because everybody GPS to buy one so I can see it is two weeks

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and GB's X now to explore X is three x three X is equal to X, which is the altitude.

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Right.

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Therefore I can see that two X that is agea is equal to each divided by three in the two, that's two

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by three into X.

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And remember AGEA is the sock'em radius.

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Right, because there's also the Sarkhan Center.

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And remember, if you joined the circle center to a vertex and that distance is called the Sarcone radius.

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All right.

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Let me make some space over here and we proceed.

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So we have seen that the Serkan radius in an equilateral triangle will be equal to two by three X,

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and we already know it is Route three by two.

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So let's not combine these two.

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If it's required in a question, you can combine it like you can write it as two by three in the root,

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three by two.

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All right.

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Now let's proceed.

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We can see that DP's X and that would be one by three.

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It's right the same manner.

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Three X is equal to each.

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So X is equal to X by three.

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So X by three is the in radius.

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Remember, if you take the in center and you take this length over here, because when you draw an inner

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circle, all the sides are tandem's.

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Right.

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Therefore this distance over here will be the in radius and the radius in an equal triangle is equal

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to one by three X.

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So we have learned a lot of interesting properties about the equilateral triangle.

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Let's quickly revise the first properties that the median angled Bicego.

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All the different perpendicular by sector are all the same line.

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Secondly, we have seen that the center satcom center centroid, an auto center is the same point and

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we have seen that the circle radius of an equal triangle is equal to two by three each.

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And we have seen that the radius of an equal triangle is equal to one by three.

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And you know the logic behind this, right?

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So it's very easy for you to remember.

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And we collect and apply in questions because you're not just building it, but you know why the circumference

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is equal to by three X and you know why the radius is equal to one by three.

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It's in an equilateral triangle.

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And remember, in an equilateral triangle or these four points coincide.