HYACINTHOS 2768

Antreas P. Hatzipolakis

 

Let ABC be a triangle and A'B'C' the pedal triangle of a point P = (x:y:z)
in normals.

For which points P we have:

1. AB' + AC' = BC' + BA' = CA' + CB'

2. AB' + AC' BC' + BA' CA' + CB'
-------- -------- -------- ??
BC CA AB


Let Ab, Ac be the orth. proj. of B', C' on the bisector of angle A
Bc, Ba C', A' B
Ca, Cb A', B' C.


A
/|\
/ | \
/ Ab B'
/ | \
C' Ac \
/ \
/ P \
/ \
B----A'-----------C

For which points P we have:

3. AAb + AAc = BBc + BBa = CCa + CCb

AAb + AAc BBc + BBa CCa + CCb
4. --------- = --------- = ---------
BC CA AB

5. B'Ab + C'Ac = C'Bc + A'Ba = A'Ca + B'Cb

B'Ab + C'Ac C'Bc + A'Ba A'Ca + B'Cb
6. ----------- = ----------- = ----------- ??
BC CA AB


Let A'b, A'c be the orth. proj. of B', C' on the bisector of angle B'PC'
B'c, B'a C', A' C'PA'
C'a, C'b A', B' A'PB'.


A
/ \
/ \
/ A'b B'
/ / \
C' A'c \
/ / \
/ P \
/ \
B----A'-----------C

For which points P we have:

7. PA'b + PA'c = PB'c + PB'a = PC'a + PC'b

PA'b + PA'c PB'c + PB'a PC'a + PC'b
8. ----------- = ----------- = -------------
BC CA AB

9. B'A'b + C'A'c = C'B'c + A'B'a = A'C'a + B'C'b

B'A'b + C'A'c C'B'c + A'B'a A'C'a + B'C'b
10. ------------- = ------------- = ------------- ??
BC CA AB



Answers:

1. P = ( - tan(A/2) + tan(B/2) + tan(C/2) ::)

2. P = ( - sin^(A/2) + sin^2(B/2) + sin^2(C/2) ::)

sin(A/2) sin(B/2) sin(C/2)
3. P = ( - ---------- + ---------- + ---------- ::)
cos^2(A/2) cos^2(B/2) cos^2(C/2)

sin^2(A/2) sin^2(B/2) sin^2(C/2)
4. P = ( - ---------- + ----------- + ---------- ::)
cos(A/2) cos(B/2) cos(C/2)

1 1 1
5. P = ( - -------- + -------- + -------- ::)
cos(A/2) cos(B/2) cos(C/2)

6. P = ( - sin(A/2) + sin(B/2) + sin(C/2) ::)

1 1 1
7. P = ( - -------- + -------- + -------- ::)
sin(A/2) sin(B/2) sin(C/2)
1 + tan(A/4)
8. P = ( - cos(A/2) + cos(B/2) + cos(C/2) ::) = ( ------------- ::)
1 - tan(A/4)

1 1 1
9. P = ( - -------- + -------- + -------- ::)
cos(A/2) cos(B/2) cos(C/2)

10. P = ( - sin(A/2) + sin(B/2) + sin(C/2) ::)


Antreas