# 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
2 years, 5 months ago

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They are two cyclic kites meaning symmetric pair of angles are right,which directly implies the second property

- 2 years, 5 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 ?

- 2 years, 3 months ago

I m not able to understand the location of point N as intersection of BC and CP makes point C as point N

- 2 years, 5 months ago