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

Quadrilaterals $BXIM$ and $XCNI$ are kites.
$MIN$ are collinear and $XI$ is perpendicular to $MN$ .

#Geometry

Easy Math Editor

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Comments

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

Xuming Liang - 5 years, 6 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 ?

Shrihari B - 5 years, 5 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

Harmanjot Singh - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$