## Brocard-like points

Expand Messages
• Let ABC be a triangle, P a point and A B C the pedal triangle of P. AaAbAc := The Orthic Triangle of AB C (ie Ab = the Orth. Proj. of B on AC , Ac = the
Message 1 of 1 , Sep 1, 2006
Let ABC be a triangle, P a point and A'B'C'
the pedal triangle of P.

AaAbAc := The Orthic Triangle of AB'C'
(ie Ab = the Orth. Proj. of B' on AC',
Ac = the orth. proj. of C' on AB')

BaBbBc := The Orthic Triangle of A'BC'
(ie Bc = the orth. proj. of C' on BA',
Ba = the orth. proj. of A' on BC')

CaCbCc := The Orthic Triangle of A'B'C
(ie Ca = the orth. proj. of A' on CB',
Cb = the orth. proj. of B' on CA')

For which points P

(1) The Circumcircle of AcBaCb is centered at P?

(2) The Circumcircle of AbBcCa is centered at P?

If P = (x:y:z) in Trilinears, then

(1) ==> y^2 + (zsinA)^2 = z^2 + (xsinB)^2 = x^2 + (ysinC)^2

==> The squared Trilinears of P are

((sinAsinC)^2 + (cosC)^2 : (sinBsinA)^2 + (cosA)^2 :
: (sinCsinB)^2 + (cosB)^2)

(2) ==> z^2 + (ysinA)^2 = x^2 + (zsinB)^2 = y^2 + (xsinC)^2

==> The squared Trilinears of P are

((sinAsinB)^2 + (cosB)^2 : (sinBsinC)^2 + (cosC)^2 :
: (sinCsinA)^2 + (cosA)^2)

Equality of the altitudes ie
for which points P

(1) AcC' = BaA' = CbB'

(2) AbB' = BcC' = CaA'

(1) ==> P = (1 + cosAcosC - cosC : 1 + cosBcosA - cosA :
: 1 + cosCcosB - cosB)

(2) ==> P = (1 + cosAcosB - cosB : 1 + cosBcosC - cosC :
: 1 + cosCcosA - cosA)

Antreas

--
Your message has been successfully submitted and would be delivered to recipients shortly.