3

25

X(25)    PERSPECTOR OF ORTHIC 
and TANGENTIAL TRIANGLES

Triangle A'B'C' is the orthic triangle of 
ABC. Triangle A''B''C'' is the tangential 
triangle of A'B'C'. Lines A'A'', B'B'', C'C'' 
concur in X(25), labeled 25.

Sketch contributed by Michael Stairs, 
March 2018.

Drag A, B, C. 

A = (8.63179994957998, 4.685439807509034)
B = (4.8533879939985045, 0.6047548954810488)
f = Segment(A, B)
C = (10.02, 0.6)
g = Segment(B, C)
h = Segment(C, A)
i = OrthogonalLine(A, g)
j = OrthogonalLine(C, f)
k = OrthogonalLine(B, h)
C' = Intersect(j, f)
A' = Intersect(i, g)
l = Segment(C', A')
B' = Intersect(k, h)
m = Segment(A', B')
n = Segment(B', C')
p = Line(C, m)
q = Line(B, l)
r = Line(A, n)
C'' = Intersect(q, r)
B'' = Intersect(p, r)
s = Segment(C'', B'')
A'' = Intersect(p, q)
t = Segment(B'', A'')
a = Segment(A'', C'')
b = LineBisector(f)
c = LineBisector(g)
J = Intersect(b, c)
d = Circle(J, C)
e = Line(C'', C')
f_1 = Line(B'', B')
g_1 = Line(A'', A')
D = Intersect(e, f_1)
h_1 = Segment(B'', D)
i_1 = Segment(D, A'')
j_1 = Segment(C'', D)
k_1 = Segment(J, B)
l_1 = Segment(J, C)
m_1 = Segment(J, A)
n_1 = Segment(C, C')
p_1 = Segment(A, A')
q_1 = Segment(B, B')