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IMO 2022 - P4: The reasoning behind the 'easy' geometry problem
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IMO 2022 - Problem Number 4
You need only know angle chasing, concyclic quads, and congruency+similarity to do this problem.
Latex:
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.
TIMESTAMPS:
00:00 Intro 10 - 30/60 - 270
00:32 Drawing the diagram
04:13 How we'll draw
05:35 Geometry philosophy
06:38 Forwards Geometry: Idea 1
08:05 Forwards Geometry: Angle Chasing
10:08 Seeing a mistake!
10:27 Finding a mistake
11:13 Forwards Geometry: Idea 2
11:34 Backwards Geometry: Idea 1
13:02 Forwards Geometry: Angle Chasing + Idea 3
16:34 Backwards Geometry: Idea 2
18:37 Forwards Geometry: Final Idea
20:19 Reflections on the Problem
23:40 Thanks for Problem Solving :)
You need only know angle chasing, concyclic quads, and congruency+similarity to do this problem.
Latex:
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.
TIMESTAMPS:
00:00 Intro 10 - 30/60 - 270
00:32 Drawing the diagram
04:13 How we'll draw
05:35 Geometry philosophy
06:38 Forwards Geometry: Idea 1
08:05 Forwards Geometry: Angle Chasing
10:08 Seeing a mistake!
10:27 Finding a mistake
11:13 Forwards Geometry: Idea 2
11:34 Backwards Geometry: Idea 1
13:02 Forwards Geometry: Angle Chasing + Idea 3
16:34 Backwards Geometry: Idea 2
18:37 Forwards Geometry: Final Idea
20:19 Reflections on the Problem
23:40 Thanks for Problem Solving :)
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