O is the circumcenter of acute triangle ABC, H is the Orthocenter. AD is perpendicular to BC, EF is the perpendicular bisector of AO,D,E on the BC. Prove that the circumcircle of triangle ADE passes through the midpoint of OH.

(In reply to Some thoughts by Bractals)

Using Bractal's suggested points except B=(b,0) [I find it easier to let b be negative] the algebra I will spare you but it did fit on one side of a page:

H = (0,-bc/a)
O = ((b+c)/2,(bc+a²)/(2a))
M = ((b+c)/4,(a²-bc)/(4a))

E = ((a²b²+a²c²+b²c²+4a²bc-3a^4)/(4a²(b+c)),0)

P = ((a²b²+a²c²+b²c²+4a²bc-3a^4)/(8a²(b+c)),a/2)

Then

|PM|² = |PD|² =
((a²b²+a²c²+b²c²+4a²bc-3a^4)²+16(b+c)a^6)/(64(b+c)²a^4)
yes, they did both come out to this.

Posted by Jer on 2013-02-04 13:48:42