Orthocenter2

Orthocenter's further properties

Continuing the discussion started in Orthocenter.html :
1) The circles c_CH, c_AB with diameters CH and AB are orthogonal.
2) The products AH*HE = BH*HF = CH*HD.
3) H is the radical center of any three circles passing through points (A,E), (B,F), (C,D) respectively.
4) angle(D'CC') = |angle(B)-angle(C)|.
5) angle(AJE) = 2angle(B). Triangle AJE is isosceles.
6) Let L be the intersection point of c_CH, c_AB, then LAJE is a cyclic quadrilateral.
7) Points L, H, J, C' are on a line, which bisects angle(ALE).
8) The symmetric P of D' w.r. to O, A, K and the intersection M of circumcircle of LAJE with BC are on a line.
9) CABP is an isosceles trapezium.
10) The Simson line P'P'' of P is parallel to CC'.
11) The orthocenter H and the circumcenter O are isogonal conjugate.
12) Triangle IKO is isosceles and angle(I)=angle(O) = angle(B).
13) The angle of lines e and KM is equal to angle(KIH).
Exercise: express the angle HC'D' in terms of triangle's data.

1) Follows by measuring the angles of triangle IJE.
2) Follows from the cyclicity of quadrilaterals FABE, BDFC, etc.
3) Follows from 2.
4) Follows by measuring the angles of triangle CDC'.
5) AJE is central to B. Note that IJ is parallel and half of CC' and apply 4.
6) Compare the opposite angles at J and L.
7) Follows from 6.
8) Measure angles at M.
9) Obvious.
10) Measure the angles at P' and B using 9.
11) Follows from the equality of angles angle(ACD)=angle(C'CB).
12) IO is parallel and half of HC'. The medial line of LJ must pass through the middle of IO. Besides LM, KO are both orthogonal to LA, hence parallel and angle(KOI) = angle(JLM) = angle(B).

Orthocenter's further properties

See Also