Waste less time on Facebook — follow Brilliant.
×

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, 1 month ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

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

Log in to reply

@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 · 11 months ago

Log in to reply

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 · 1 year ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...