Common Orthocenter and Centroid Showing that a triangle with the same point as the orthocenter and centroid is equilateral
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- We're asked to prove that if the orthocenter and centroid
- Of a given triangle are the same point
- Then the triangle is equilateral
- So I have a triangle over here and we're going to assume
- That its orthocenter and its centroid are the same point
- And just so to review the orthocenter is the point where
- The three altitude of a triangle intersect
- And the centroid is the point where the three medians
- So we can do it is we could assume
- these three lines right over here
- that these are both altitude and median
- And that this point right over here
- is both the orthocenter and the centroid
- So if we assume that these lines are altitudes
- And that tells us that they are perpendicular to the opposite sides
- So that is a 90 degree angles those are both 90 degree angles
- These are both 90 degree angles
- These are both 90 degree angles
- And the fact that it's a centroid means
- That each of these lines bisect the opposite side
- So it tells us that this length is equal to length
- That this length let me just use a different color
- This length is equal to that length
- And it tells us that this length this length is equal to that length
- And you might also say
- Well gee each of these line are also then the perpendicular
- Bisectors of each of the sides
- So not only is this the orthocenter in the centroid
- It is also the circumcenter of this triangle right over here
- For that out of the way we kind of
- Marked up everything that we can assume given that
- This is an orthrocenter and a center
- Although the other things other properties of this
- Especially centroids that we know
- But now let's prove that this has to be an equilateral triangle
- So the first thing that you might see
- And let me label some letters here
- So we can refer to things a little bit better
- So let's call that A B C D E and F
- And we could label in the centroid right over here G
- So the first thing let's look at triangle AFG
- And triangle this is E and triangle EFG
- So let's compare triangle AF triangle AFG
- Let's compare that to triangle EFG triangle EFG
- So they definitely have one side
- Side EF is congruent to side AF these are both the same
- And we have angle EFG as the same as angle AFG
- They're both 90 degrees
- And then they both clearly share the side FG
- They both share this side right over here
- So they have a congruent side an angle
- A corresponding angle in between in another congruent side
- And they're all so congruent side congruent corresponding angle
- And another congruent corresponding side
- So by side angle side these two characters
- Are going to be congruent by side angle side congruency
- Now can you make the exact same argument
- We can make the exact same argument to say
- That all of these pairs that have the kind of share this
- That both have these, these 90 degree angles next to each other
- Are going to be congruent
- Triangle we can by the same exact argument
- We can see triangle EDG triangle EDG
- Is going to be congruent to triangle CDG
- Triangle CDG same exact thing side angle
- And then we have a side right over here
- And then we can use the exact same argument
- For this one right over here triangle C looks like an A
- Triangle CBG is congruent to triangle ABG
- So that by itself is interesting
- But we know that if two triangles are congruent
- All of their corresponding sides and angles
- Are going to be congruent
- So for example if we know
- That the measure of this angle is blue
- The corresponding angle on this triangle
- Is also going to have the same measure
- I'll just make it with that same blue angle
- And we know if this angle right over here is magenta
- The corresponding angle on triangle AFG
- Is also going to have that same measure
- And I'll just mark it with that magenta again
- Now we also know from our properties of vertical angles
- That whatever an angle measure of AFG is
- DGC is going to have the same measure
- Because they are vertical angles
- But we know whatever angle measure this is
- This triangle, triangle CDG is congruent to triangle EDG
- So corresponding angles have to be congruent
- So this angle is this magenta measure
- Then this angle also has to be the magenta measure
- And then once again you can see vertical angles
- That this is magenta then this is also gonna have
- That same measure
- And if this has a measure then
- That is also going to have that same measure
- So by using a little bit of argument of congruent triangles
- Corresponding angles are going to be congruent
- And vertical angles we can see that all of these inner angles
- Right over here are gonna have to be the same measure
- They're all and I'm using that with this little magenta arc
- Right over there
- Now all of these triangles if we split this triangle in two
- They all have a 90 degree angle they all have a magenta arc
- So whatever's left over is going to be
- 180 minus 90 minus magenta
- Or it's really 90 minus this magenta angle right over here
- And that's what this blue angle must be
- This blue angle is essentially 90 minus the magenta angle
- The blue angle is 90 minus the magenta angle
- And so that blue angle must be the third angle for all of these
- So once again this blue angle is going to be
- 90 minus magenta angle
- Or a 180 minus magenta minus the 90
- So that's gonna be the blue angle
- If you, essentially what we're saying is
- If you know two angles of a triangle
- that forces with the other triangle
- What the other triangle is going to be
- We already know that all 6 of these triangles
- Have two angles in common
- The 90 degree angle and the magenta angle
- So they all the third angle must be the same as well
- And we're specifying that with this blue angle
- This blue angle right over here
- And now we see if we're to look at angle
- If we we're to look at angle E EAC
- So angle EAC we see that that's just two of these blue angles
- That is congruent to angle ACE, ACE
- Which just two of those blue angles
- Which is congruent to angles CEA, angles CEA
- Which is once again just two of those blue angles
- So we have a triangle here where the three angles
- In that triangle all are congruent
- So it is an equilateral triangle
- It is a 60 degree we've proven before
- If all three of your angles are the same
- The lengths of all three sides are the same as well
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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