Comparing Incircles

Geometry Level 4

A triangle \(ABC\) has incentre \(I\). Consider lines \(l_a\) passing through \(A\), \(l_b\) passing through \(B\) and \(l_c\) passing through \(C\) and respectively parallel to sides \(BC\), \(CA\) and \(AB\). By the intersections of these lines, we obtain a triangle \(\Delta_1\).

Now, reflect the line \(l_a\) across line \(AI\) to obtain line \(L_a\), reflect the line \(l_b\) across line \(BI\) to obtain line \(L_b\) and reflect the line \(l_c\) across line \(CI\) to obtain line \(L_c\). By the intersections of these new lines, we obtain a triangle \(\Delta_2\).

Then the correct relationship is:

(* Definitions: The reflection of a point \(P\) across a line \(l\) is another point \(P'\) such that \(l\) is the perpendicular bisector of segment \(PP'\). The reflection of a line \(l_x\) across a line \(l\) is another line \(L_x\) such that the reflection of each point of \(l_x\) across line \(l\) lies on \(L_x\).)

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