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Discovered Geometry Properties

Given a non-isosceles triangle \(ABC\) with incenter \(I\) and circumcircle \(\omega\). Denote the midpoints of arcs \(BC, AC, AB\) that does not contain the opposite vertex by \(X,Y,Z\) respectively. Denote \(P\) the midpoint of arc \(BC\) containing \(A\). Denote the intersection of \(BP\) and \(ZX\) as \(M\) and the intersection of \(XY\) and \(CP\) as \(N\). Prove that

  1. Quadrilaterals \(BXIM\) and \(XCNI\) are kites.

  2. \(MIN\) are collinear and \(XI\) is perpendicular to \(MN\).

Note by Alan Yan
10 months, 3 weeks ago

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They are two cyclic kites meaning symmetric pair of angles are right,which directly implies the second property Xuming Liang · 10 months, 2 weeks ago

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@Xuming Liang Yea even i did it that way ... although in the second part i could first prove the second statement and then the collinearity. This was a nice geometry problem @Alan Yan Can you also post some more geometry problems ? Shrihari B · 8 months, 3 weeks ago

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I m not able to understand the location of point N as intersection of BC and CP makes point C as point N Harmanjot Singh · 10 months, 1 week ago

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