TriangleSimilaritiesProd

Product of triangle-similarities

Given is the triangle ABC, its Apollonian circle (EC) (locus of points C* s.t. C*A/C*B = CA/CB), its circumcircle (DA) and circle (GA) passing through A, B, D. Points C, H, J (J diametral of D w.r. to G) are on a line. Point I, intersection of lines ED and CH, defines triangles s=(AIH) and r=(IHB), similar to the original one. The crucial fact for the proof (collinearity of point C, H, J) is contained in the file: Apollonian_rel.html .

Using the previous facts one can easily calculate the product of the three similarities:
U = U(A, AB/AC) with center at Į, rotation-angle (BAC) and ratio AB/AC,
V = V(B, BC/AB) with center at B, rotation-angle (CBA) and ratio BC/BA,
W= W(C,CA/CB) with center at C, rotation-angle (ACB) and ratio CA/CB.
Then U(H)=I, then V(I) = H. Hence the composition of these two similarities R = V*U leaves H fixed. One verifies also that U(B) = C and V(C) = A. Thus R(B) = A. Hence R is the spiral similarity with center H, rotation-angle (BHA) = (BAC)+(CBA) and ratio HB/HA = CB/CA.
Now W(A) = B. Hence the composition: Ō = W*R = W*V*U, will have T(B) = W(R(B)) = B (fixed). Hence it will coincide with a similarity at Ā, with rotation-angle pi and ratio the product of the ratios of R and W. This product is 1, hence T will coincide with the symmetry with respect to B.

Product of triangle-similarities

See Also