All Hail Euler the ruler!
Consider a triangle Δ A B C whose circumcenter is at the origin. If in Δ A B C , the coordinates of the centroid G are ( x G , y G ) and the coordinates of the orthocenter H are ( x H , y H ) .
Find the ratio x G y G x H y H .
Bonus : Don't forget the lonely incenter!
The answer is 9.000.
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2 solutions
Let C = ( x C , y C ) = ( 0 , 0 ) .
We know that G O G H = 1 2 .So we will use section formula , where m = 2 , n = 1 .
( x G , y G ) = ( m + n m x C + n x H , m + n m y C + n y H ) = ( 3 x H , 3 y H ) … s i n c e ( x C , y C ) = ( 0 , 0 ) ⇒ r a t i o = x G y G x H y H = ( 3 x H ) ( 3 y H ) x H y H = 9
Bonus: Similarly using section formula , we can find the coordinates of incenter too.Try it!
OGH is a st. line. We know OG:OH::1:3.... O(0,0). Let G(m, n) then H(3m, 3n).
So X G ∗ Y G X H ∗ Y H = m ∗ n 3 m ∗ 3 n = 9