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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
1 year, 10 months ago

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They are two cyclic kites meaning symmetric pair of angles are right,which directly implies the second property · 1 year, 10 months ago

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 ? · 1 year, 8 months ago