Hypotheses equations

Symmetric of xy respect to side OOab, take a perpendicular line to such side through xy

a(L-x)+b(M-y)=0

let LM be the intersection with the given side,-bL+aM=0. Then the symmetrical is the point (X,Y) such that (x,y)+2((L,M)-(x,y))=(X,Y).

For the other side we take RS as intermediate point, thus we have the following equations:

(a-l)(R-x)+b(S-y)
-b(R-l)+(a-l)S
Z-x-2(R-x)
W-y-2(S-y)
blh-1


where the last one avoids degenerate cases such as b=0 or l=0.

Step 2. Thesis
(XW-Zy+xY+yX-YZ-xW)=0

Step 3. Checking the conjecture.
NormalForm(1, Ideal(a(L-x)+b(M-y),
-bL+aM,
X-x-2(L-x),
Y-y-2(M-y),
(a-l)(R-x)+b(S-y),
-b(R-l)+(a-l)S,
Z-x-2(R-x),
W-y-2(S-y),blh-1,
(XW-Zy+xY+yX-YZ-xW)t-1))
1

Thus it is false in general that the three points are aligned.

(continues in Next Page)


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