perplexus.info :: Geometry : Decagonal Intersection Delight

Decagonal Intersection Delight (Posted on 2024-04-22)

You are given a regular decagon A₁A₂... A₁₀. Let A be the intersection point of lines A₁A₄ and A₂A₅, B is the point of intersection of lines A₁A₆ and A₂A₇, and C is the point of intersection of lines A₁A₉ and A₂A₁₀. Find the angle of ABC.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 1 of 2

A1A6 and A2A7 each connect points 5 apart so they are diameters of the circumcircle of the decagon and B is the center of the polygon.

A1A4 and A2A5 each connect points 3 apart, so by symmetry, the ray BA contains point A3

A1A9 and A2A10 each connect points 2 apart, so by symmetry, the ray AC bisects the side A1A10.

The angle sought is 2.5*36=90degrees.

Posted by Jer on 2024-04-22 11:33:34

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