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Omni-MATH / Let A₁B₁C₁D₁ be an arbitrary convex quadrilateral. P is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that ed…

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Let A1B1C1D1A_1B_1C_1D_1 be an arbitrary convex quadrilateral. PP is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point PP is acute. We recursively define points Ak,Bk,Ck,DkA_k,B_k,C_k,D_k symmetric to PP with respect to lines Ak−1Bk−1,Bk−1Ck−1,Ck−1Dk−1,Dk−1Ak−1A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1} respectively for k≥2k\ge 2. Consider the sequence of quadrilaterals AiBiCiDiA_iB_iC_iD_i. i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not? ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
Plain-text mathematical notation (without MathML)
Let A₁B₁C₁D₁ be an arbitrary convex quadrilateral. P is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point P is acute. We recursively define points A_(k),B_(k),C_(k),D_(k) symmetric to P with respect to lines A_(k−1)B_(k−1),B_(k−1)C_(k−1),C_(k−1)D_(k−1),D_(k−1)A_(k−1) respectively for k≥2.
Consider the sequence of quadrilaterals A_(i)B_(i)C_(i)D_(i).
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
Original LaTeX notation
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.
Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?

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task_id
26673590-0da2-5a98-a48d-384fc82273d1
task_key
test--26673590-0da2-5a98-a48d-384fc82273d1
task_revision_id
1
config / split
/ test

Official source

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