Hi there, I've got 2 questions that I have no clue how to do.
QUESTION 1:
Prove that (AH)^2 + (BC)^2 = 4(AO)^2 where
A,B,C are the sides of the triangle
H is the orthocenter and
O is the circumcenter.
So far the only thing I've been able to determine is that AO = BO = CO
And that AH = 2OA' where A' is the midpoint of BC.
I have no clue where to go from there.
QUESTION 2:
A circle intersects sides BC, CA, AB of triangle ABC in points L and L', M and M', N and N'. Show that AL, BM, CN concur iff AL', BM', CN' concur.
Clearly this is somehow an application of Ceva's Theorem and I suspect that I have to show that L', M', N' are isometric conjugates of L, M, and N respectively, but I have no idea how to start or reach this conclusion.
PLEASE HELP!!!!!
QUESTION 1:
Prove that (AH)^2 + (BC)^2 = 4(AO)^2 where
A,B,C are the sides of the triangle
H is the orthocenter and
O is the circumcenter.
So far the only thing I've been able to determine is that AO = BO = CO
And that AH = 2OA' where A' is the midpoint of BC.
I have no clue where to go from there.
QUESTION 2:
A circle intersects sides BC, CA, AB of triangle ABC in points L and L', M and M', N and N'. Show that AL, BM, CN concur iff AL', BM', CN' concur.
Clearly this is somehow an application of Ceva's Theorem and I suspect that I have to show that L', M', N' are isometric conjugates of L, M, and N respectively, but I have no idea how to start or reach this conclusion.
PLEASE HELP!!!!!