HYACINTHOS 28843

[Tran Quang Hung]: 

 

Let ABC be a triangle with orthocenter H.

HaHbHc is pedal triangle of H.

MaMbMc is medial triangle.

Ea, Eb, Ec are midpoints of HA, HB, HC (Euler points)

 

Then

 

a) The circle is tangent to (Ha,HaH), (Hb,HbH), (Hc,HcH) which is tangent to (ABC).

b) The circle is tangent to (Ea,EaH), (Eb,EbH), (Ec,EcH) which is tangent to (ABC).

c) The circle is tangent to (Ma,MaH), (Mb,MbH), (Mc,McH) which is tangent to (ABC).

 

Which are these tangent  points?

 

Are there any triple points on NCP which have this property?

 

[César Lozada]:

a)       X(915)

b)      X(108)

c)       Tangency is false

 

 

[Antreas P. Hatzipolakis]:

 

Which points are the centers of the circles ?


[César Lozada]: 

 

a)       Assume ABC is acute(*).  The center of the circle (Ho) tangent to (Ha, HaH), (Hb, HbH), (Hc, HcH) has radius and center:

 

radius = r*(4*R^2-SW)/(R^2-2*R*r-r^2),

 

Ho = X(3)X(915) ∩ X(4)X(8)

= (a^6-(b+c)^2*a^4-(b^2+c^2)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2*(b-c)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : : (barys)
= SB*SC*(b*c+2*R^2-SW) : : (barys)
= lies on these lines: {3, 915}, {4, 8}, {11, 7040}, {24, 278}, {52, 5905}, {68, 2994}, {242, 3542}, {281, 1594}, {513, 5553}, {1068, 11398}, {1118, 1870}, {2973, 17170}, {3147, 17923}, {7510, 11396}, {10018, 17917}, {17516, 21664}

= [ 5.5539527344656890, 4.2225112465598160, -0.1854545994780572 ] // (For ABC with  a=9, b=10, c=13)

 

ETC points on (Ho): 915.

 

(*) Otherwise, the expressions require absolute values, because the radius of circles have absolute values. (Example: Radius of circle (Ha,HaH) is 2*R*|cos(B)*cos(C)|) . For ABC obtuse, the center becomes very complicated.

 

b)      Assume ABC is acute.  The center of the circle (Eo) tangent to (Ea, EaH), (Eb, EbH), (Ec, EcH) has radius and center:

Radius = r*|(4*R^2-SW)/(4*R^2-SW+r^2)|

Eo = X(7952)

 

César Lozada